The flour tortillas used to make burritos aren't usually "sealed"... they're just wrapped (actually wrapped and overlapped) so that they stay pretty well closed around the filling.

To wrap successfully so that burritos stay closed:

...don't put too much filling in each tortilla (filling means all the stuff you want in the burrito)
...lay the filling out on a flat tortilla so that it's kind of in a strip shape (not a circle, or rectangle, etc.) near but not at one end... sort of like this:

http://becheap.wordpress.com/2009/03/06/cheap-bulk-bean-burritos

...don't put much if any filling on the tortilla near the 2 outer ends of the strip (of filling)
...fold one side of the tortilla over the strip of filling and snug it down a bit
.....roll the tortilla over (half a roll)
.....THEN fold in each end of tortilla left empty at each end of the filling, back toward the center
.....then roll the burrito all the way up with the remaining tortilla

HOWEVER... here are some slightly different ways to wrap, and it looks like the **wrap both sides in first** is the favorite!:

http://bonzaiaphrodite.com/2009/07/recipe-i-heart-burrito/

http://www.thesimpledollar.com/2009/02/20/bulk-breakfast-burritos-convenient-cheap-healthy-and-easier-than-you-think

http://www.sutree.com/how-to/14670/How-to-Make-and-Roll-a-Burrito

If you use a large enough tortilla ...and don't use too much filling...and wrap it the right way, the burrito should stay closed pretty well.

TIPS:

If you aren't eating the burrito right away, you can wrap each burrito tightly in plastic wrap and refrigerate a while... that will also make everything stick together well. That's also done with all kinds of "roulades" or rolled up meats/fish with stuffings, etc. If you're cooking with flour or corn tortilla "wraps," just lay them seam-side down to prevent leakage.
If you want to eat right away, just wrap and/or enclose most of the bottom part of the burrito in foil or deli wrap, etc., to catch any leakage.

Flour tortillas can be a little stiff sometimes to wrap really well even right out of the package. Most people probably don't prepare their tortillas before wrapping with them, but "steaming" them briefly in some kind of way will make them really soft and drapable... the moisture introduced into them can make them a bit stickier from the starch too so may even "seal" to each other better.
The easiest way to steam tortillas would be to wrap (rolled up if you want) inside a piece of plastic wrap, deli wrap, damp paper towel, etc., then microwave for just a few seconds.
To "really" steam them though, put them in a steamer of some kind (bamboo steamer, metal collapsible steamer, etc) over a pot of simmering water. Or you can make your own steamer by just putting a plate inside a skillet of simmering water (held up out of the water though, by aluminum foil wads or something else), then putting a lid on the skillet to let them steam build up around the tortillas.

HTH,

Diane B.