Can one get into trouble if one doesn't show up at the gate 30 minutes before departure (or whatever time window the boarding pass is indicating)?

Lately there are 3 things to watch out for.

1. Some airlines are now auditing carry on luggage, and tagging bags.
   No tag, no board. This is over and above calling for volunteers to
   check carry on luggage. They weigh and tag at the gate and need time
   for this. Don’t turn up as the door closes.

2. It still happens that an earlier cancellelation will spill passengers onto your flight. Sometimes the first you will hear of a problem is when your validly issued boarding pass is rejected. Always check in as early as possible, and never check in at the gate, for any reason. We also hear of some aircraft being four seats short because the toilet is larger etc, and seats are allocated for a different fitout.

3. For logistical reasons (Aerobridge, stairs, spare parts), your gate may change, the new gate may be a fair hike, and not be familiar to you. Worse, if the flight is cancelled and it is a shoestring budget flight, you may not get compensation if you are not at the gate at the 30 minute cutoff. In Australia, Tiger Air are red hot on cutoff times, consequently the PAX are extremely well behaved as far as times go, very orderly and very efficient embarcation process results.

You are getting some decent response from other people, so I will cut it short based on my experience, but I will give you some exact numbers:

Even if you are 5 minutes late for the gate, you MIGHT be able to get on board. In my experience flights usually wait a few minutes for missing passengers, but normally you should be there in time, because waiting is not guaranteed.

What you MUST comply with is the following rules (subject to change based on continent, I live in Europe):

For national flights, e.g. from Aarhus to Copenhagen, which are both in Denmark you are expected to have checked in at the counters at least 45 minutes before the flight. This is so they can proccess you luggage and register you for the flight.

For international flights this time is doubled – 90 minutes (1 and a half hours) before the flight for the same reason.

I have never missed a national flight, but this one time I was flying from Istanbul (Turkey) to Eindhoven (The Netherlands). I was at the airport 3 hours before the flight, but due to long long lines I was unable to check my luggage on time. I had 1 hour and 20 minutes until my flight, but since I did not fit within the 1 hour 30 min rule I was refused check-in, and I missed my flight because this. This is also considered my fault and reimbursement of my ticket or it’s value was not provided, because it was my decision to come this late. Or in other words – long queue is something that the airport expects that I foresee.

Having said that, the first check in is your biggest trouble. If you get through that you are pretty much on the safe side.

Be sure to check the national and international flight rules for your country/continent, as they might be different, it only takes a minute of googling.

As with other answers, I will say that the real “trouble” you can get into is limited to missing your flight and all the complications that can result from not being on the flight. Once the doors are closed, it’s nearly impossible to get on the airplane (i.e. to get them to open the doors again). Not being there 30 minutes early also limits your ability to hear announcements about the flight and/or take advantage of situations which exist (e.g. on a lightly booked flight you may miss changing seats to an unoccupied row, as people who were there early may have already done so, with none left for you; on an overbooked flight you won’t be able to volunteer for additional compensation to be bumped to the next flight).

In addition, when the flight is overbooked, passengers who are not present may be bumped without volunteering. When exactly that happens is, usually, up to the gate attendants. If you are not present when it happens, you will have no say in the matter. Usually, this won’t affect you, but you have no input into the situation, let alone control of it, if you are not present.

What’s printed on your ticket/boarding pass is what the airline is agreeing to as part of your purchase. If you don’t comply, then, at a minimum, you have much less standing when requesting resolution after missing the flight. In fact, for non-refundable tickets, it’s likely the airline will take the position that you didn’t live up to your side of the contract and provide no compensation or rebooking (i.e. you’ll have to purchase a new ticket for a later flight at the the currently available rates). What actually happens can largely depend on your relationship with the airline (e.g. frequent flier status) and how the agent at the gate/booking desk feels about you (i.e. be nice, polite, and ask for assistance; you can always get pissed-off later).

Do gates close earlier than what is stated on the boarding pass?

In comments, you’ve asked multiple times “do gates close earlier than what is stated on the boarding pass”. The answer is that airline personnel usually wait until after the printed time to close the doors. However, yes, sometimes the doors are closed early. When the doors actually get closed is up to the flight crew and gate attendants.

My personal experience is that it’s possible to miss your flight, even if you arrive 30 minutes early. I’ve had one flight where I arrived more than 30 minutes prior to the scheduled departure, but the plane had already departed. This happened with a major airline, at a major airport, but with a small airplane. The airline acknowledged the plane departed early, but gave no explanation as to why (or at least I don’t recall them giving an explanation). They did re-book me to the next available flight, which was the next day at the same time. Needless to say, I arrived quite early the next day.

Theoretically you can miss your flight.
Practically… It’s another story.
I’ve flown quite a bit, locally in an European country and overseas.
Given the vast amount of nervously screaming flight attendants picking up passengers even before security and passing them through quick lanes or just past all the other people who have the one more braincell to read the clock on every airport and every flight I’ve been on in at least the last 10 years. You’re likely to get your plane if you’re still at the gate before! departure.

I once had a very badly printed receipt making me think the boarding time was actually the departure time. So I showed up at that time and when I reached the counter of the airline and she phoned the plane they just had closed the doors. If not she would have halted the process and quick pass me through everything. And that was a low cost airline!

Still… I’d just be on time, you never know what will happen.

You could be stuck in some in between check depending on which airport you are in and where you fly to for example. If you need to go through immigration could be one case. I don’t think they are in any way obliged to treat you with priority if you are late. Still I have seen flight attendants drag those people through there and the officers play along. Probably you give up on that after some time or whatever.
Just one other example is when you need to be brought to the plane from the gate, for instance with a bus. If they ordered three buses and all are gone they probably close the gate. There’s actually many possibilities.

Update

Seems I’m right with this:

My guess is the hard limit of closing the doors is because after this the pilot confirms the take off window to the airport. Maybe it costs extra if you then change the window again. Until confirmed you probably are just automatically passed through the line of waiting planes and that doesn’t cost. So just being late INSIDE the take off window is no problem for the airline.

But it’s not about fees but more like time is money.

If the airplane isn’t ready for departure, we risk missing the slot time, and thereby causing delay. This is the reason why passengers need to be seated well ahead of the slot time. We normally also send a message to air traffic control, stating that we are ready (all passengers onboard, all doors and hatches closed), hoping for an earlier slot time.

Found in:
https://scandinaviantraveler.com/en/aviation/ask-the-pilot-what-is-a-slot-time
The same page states the slots are rather short -5/+10 minutes:

Calculated Take-Off Time (CTOT). The CTOT is the time, with a window of -5/+10 minutes, when the airplane is required to be at the runway, ready for departure. The slot time controls other standardized events, such as when the airplane can leave the gate (Target Off Block Time, TOBT)

It’s so you don’t miss relevant announcements for your flight, as in most large airports these can only be heard in the vicinity of the gate.

This can include things like a delay in boarding, or change of gate. If a flight is overbooked, they will ask for volunteers to take a later flight for compensation, or they may ask for volunteers to check their carryons if they don’t expect to have enough overheard storage. In rare circumstance they may have to reassign seating and reprint boarding passes (say, if they had to use a plane with a different layout at the last minute). Sometimes upgrades will happen at the gate shortly prior to boarding. I was once on a flight where they used the time leading up to boarding to weigh everyone’s carryon, since there was a surcharge for excess weight. Sometimes there is extra security, as mentioned by @jpatokal.

Anyway, those are a few examples that I’ve experienced of why you may want to be at the gate prior to boarding.

The deadline that matters is when the gate closes. If you’re there 1 minute before closure, you’re fine; if 1 minute after, you’re going to have a bad time. The 30 minutes is there mostly for the airline’s benefit, since they’d prefer to have you there early.

One caveat is that at airports with security checks at the gate (Singapore Changi, for example), you will want to get there a few minutes earlier to make sure you have enough time to get through security.

The only trouble you can get into is a) missing the flight, and b) if the flight has following segments (or a return) on a multi-flight ticket, having the remaining other segments cancelled without refund.

As @Patricia Shanahan notes below, the later you line up to board, the more likely the bins will be full and you’ll have to gate-check your bag.