Is there a risk to write an invitation letter for a stranger to obtain a Czech (Schengen) visa?

In Czech Republic, invitations (pozvání) are highly formalized and often impractical. Here’s the process:

1. Depending on the type of invitation (see below), you may need to prepare a bank statement or ATM slip showing how much funds you have.
2. You visit a foreigners’ police office (odbor cizinecké policie) and fill out a form on a computer. The form is printed out and signed by you, then it remains with the police for certification. You get a date for the next visit (typically about 5 working days later).
3. You visit the police office again, bringing 300 Kč in fee stamps (kolky) to collect the certified invitation form.
4. You send the form to your visitor by mail or delivery service. Note that international mail may be crazily slow.
5. Your visitor goes to the embassy/consulate/visa center with the invitation form (and other prerequisites) and files a visa application.
6. Your visitor collects the visa.
7. Your visitor arrives.

Prior to his/her arrival, you may revoke the invitation at any time, and his/her visa will also be revoked. However, you cannot disclaim responsibilities you had assumed while he/she is in the ČR.

There are two possibilities with invitations:

1. You may assume all costs related with the visit. In this case you need to show that you have enough financial means available to you. For European countries (and Armenia counts as an European country), that’s at least 20000 Kč + 1100 Kč per day, full rules here.
2. You may assume only costs related to accommodation. If you invite your visitor to stay in your own place, you don’t need to demonstrate any money. In this case, the visitor will need to provide his own bank statement (and possibly air/train/bus tickets) when applying for the visa to demonstrate that he/she has enough funds for the visit.

Another possibility for your visitor is to book hotel(s) and bring hotel voucher(s) (vouchers from sites such as agoda.com suffice) instead of an invitation letter when applying for a visa. In this case, he/she will still have to demonstrate enough funds for transportation, accommodation and subsistence. However, it is important to know that the Czech Republic is particularly notorious for revoking visas issued to visitors who cancel their hotel bookings afterwards.

In any case, medical insurance is one of the prerequisites for visit visa, so your friend will have to get one before applying.

There is no requirement that visitors to the Schengen area (which Czechia is part of) need to be invited by a citizen or resident.

The standard procedure is that your friend would apply for a visa for a tourist visit on his own behalf and get a visa on the strength of his own circumstances. Then once he arrives he’d be free to visit you or not.

Generally you would only be involved in the visa application if he cannot afford the trip himself and you are good enough friends that you’re planning to help with his expenses or to let hims stay in your home. In that case you would need to supply a letter to the consulate that confirms you intend to provide that support, which will help them judge if his budget makes sense. Generally speaking writing such a letter would not give you any legal responsibility (as long as you’re truthful in describing what your intentions are).

If he can afford including a hotel in his budget for the days he would be visiting you, it would make his visa application both more straightforward and more likely to succeed to do it that way. That doesn’t mean he has to actually stay at that hotel; if he’s worried about promising something he might not keep, he could still include hotel expenses in the budget and put a footnote like “for the three days in Prague possibly stay with an online friend instead; arrangements are not finalized yet”.

It is also possible, if he mentions meeting up with you as part of his description of his travel plan, that he would like to include a letter from you simply as evidence that he’s not making it all up. In that case you can consider writing a short letter that states just the facts — how you know each other, how long time you’ve known him, and that you’re open to meet up at roughly such-and-such time and location for a few hours/days/whatever. Don’t attempt to serve as a character witness for him or sound like you’re recommending that he get a visa (the consulate won’t care about your recommendation anyway).

Some countries also have some different and more formal way for you to claim legal responsibility for some of the risk of letting your friend in (such as the German Verpflichtungserklärung) — but it doesn’t sound like you are such close friends that you ought to touch that at all. In any case, those things are only for lifting otherwise hopeless visa applications into the “barely scraping by” category. Don’t let anyone tell you they are standard for all visitors.

Your friend’s offer to send money to you sounds like a bit of a red flag. It sounds a bit too eager to get something specific from you that he shouldn’t really need if his story holds up. As far as the visa goes, you’re not the one he ought to convince about his finances; the consulate is.