I plan to visit Vilnius and hire a bike for city tour. The safety guide says that the third party liability insurance is mandatory in order to use the service. How can I purchase this kind of insurance online or in local offices in Vilnius? Is it possible for non-resident? I'm just staying there for 2 days.
To clarify things beforehand, liability insurance is not generally needed for bicycling in Lithuania (to the best of my knowledge), but to use the services of Cyclocity Vilnius. On their website it says
Make sure you have third party liability insurance, which is mandatory in order to use the service.
Liability insurance is generally very good to have so my first recommendation would be for you to get one in your home country or country of residence (you indicate Belarus) and make sure that the terms & conditions also cover your travel to Vilnius.
Otherwise you could try to purchase a liability insurance just for travel. You will likely still be better off with offers from Belarus but you could also try with international providers. The first hit googling for "liability insurance for travel" gives me this (apparently American, but targeting internationally) provider (no affiliation), I am sure there are many more upon a more thorough search.
Alternatively, I found two other providers of bike rental in Vilnius (they also offer bike tours):
They do not mention need of a liability insurance but be aware that the system is different from Cyclocity (which is similar to Velib in Paris, i.e. you can drop and take bikes at/from several stations in town) and you rent a bike e.g. for the day.
Finally - though this is not recommended and only my assumption - I do not believe that Cyclocity checks that you do have liability insurance. I would not know how to prove mine, I do not carry around my contract when traveling. And if you do get in trouble, you are in double trouble.