Do you need to take the most direct route with an Oyster Card?

Just to add what others have said when using the Oyster card there are time limits of travel after which the system will not link your "touch in" to your "touch out" and effectively charge you twice the applicable maximum fee.

How this works in practice is that whenever you touch in with your Oyster card, the system will immediately deduct the maximum travel fee from your card (this fee depends on multiple things, including whether you are on a National Rail based station or TFL based station as well as on whether it’s the on-peak or off-peak period). Whenever you touch out the system will calculate how much you travelled (including if you avoided Zone 1 by using the pink readers), and refund you the difference in your balance.

In case your travel time is longer than the maximum allowed (which depends on multple things, like how many zones you travel through as well as on whether you use National Rail services only, TFL only or both), then the system as mentioned above will not link the two trips together, so it will deduct the maximum funds from you when you touch in, then (without giving you a refund) also deduct the maximum applicable fare for the second trip when you touch out.

However even if that happens you can still get a refund. First make sure you register your Oyster card online on the TFL website, after which you’ll be able to see your trips. Once there you can see the errorneus trips, and you can ask customer service to merge/fix the problematic ones. You will likely have to describe to them what route you have taken, and they will usually apply a proper refund onto your Oyster card. Do note that since Oyster cards are "offline" in order for you to receive that refund you actually have to touch in at a station (and then do a proper, full trip to somewhere else – checking out immediately will cause extra trouble) to get that refund onto your card, so if you are leaving the London/Oyster area you could be out of luck.

Some of the issues above can be fixed by using a Contactless credit/debit card (or any other contactless payment menthod, like Apple or Google Pay) instead of an Oyster. When using these cards there are some automated systems that can automatically fix up your journey history and immediately give you a refund in case you touch in or out incorrectly or spend too much time travelling. Even if it doesn’t fix it up immediately, similarly to an Oyster card, you can also register your Contactless card online, and ask for a refund. This refund will be applied to your credit/debit card without any extra interaction, so you can even get it you have left the London area already.

Note: you should usually raise travel disputes like above on the website within two weeks.

As long as you stay within the maximum permitted journey time as described in the other answers, you can take any route you like within the Oyster network.

One additional point not mentioned in the other answers is that for journeys across London, TfL will by default assume that you have travelled through zone 1 and will therefore charge you a zone 1 fare, unless you swipe on one of the pink oyster card readers that are located in certain stations around zone 2 to prove that you took the longer way round.

As I understand it.

Oyster pay as you go does not have any concept of "permitted routes" like regular rail ticketing does.

It does however have a concept of "maximum journey times", if you spend too long on a journey, then it may time out and you may get charged for two "incomplete journeys". Incomplete journeys do not count towards daily caps.

There are also various features in the system like "continuation exits" which are designed to automatically cancel accidental multiple taps and "out of station interchanges" which join together multiple touch-in/touch-out pairs into a longer journey.

If you are using Oyster to travel from A to B via a reasonable route, then going off to do something else before making your next journey then you will almost certainly have no problems. The times are quite generous, so if you go a stop or two beyond your destination and then double back you are still probably fine.

But if you ride out to the end of a long line and back without touching out and then touch out near to where you started you are likely to fall foul of the maximum journey time rules.

If you travel to the end of the line and then touch out and back in in quick succession then unfortunately you can still have problems. At stations without barriers the "Continuation exit" feature may cause what you intended to be touching back in to accidentally be registered as a second touch out. Stations with barriers shouldn’t have that issue but you may still run into issues with out of station interchanges joining together journeys into a journey that violates maximum journey times.

If you want to spend a day riding around on trains/tubes in London then the Railforums guys recommend a paper travelcard. It’s a little more expensive than the corresponding daily cap but it avoids any issues with journey timeouts or misinterpreted touches.

There are time limits. See

https://tfl.gov.uk/fares/how-to-pay-and-where-to-buy-tickets-and-oyster/pay-as-you-go/keep-within-maximum-journey-times

As long as you stay below the maximum time for your route, you are fine. If you exceed it you may get charged twice the max fare for that zone combination.