How long can tiger barbs survive without food?

They will last longer without food than they will without clean water. A couple of weeks without food will not starve them to death but they would loose condition. A couple of weeks with no filter and no water changes would make them quite sick.

The basic rules for keeping any kind of fish sucessfully are. :- 1 inch of fish needs at least 1 gallon of water. :- Every tank must have a permanently running cycled filter. :- Every tank must have at least 50% of its water replaced every week. If you stick to the above rules your fish will stand a chance of survival. If you miss out on any of the above I can guarantee that your fish will be constantly getting sick and dying before they should.