As a disclaimer, I've only done this with very small holes in a window screen. I'm not sure if it would work well for holes greater than 1"x1". That being said, gather your supplies:
- transparent tape (scotch tape or packing tape) whose width is at least as wide as the hole
- window screen removed from window for easy access
Use a pair of scissors to cut a neatly-trimmed square or rectangle of tape that is big enough to cover the hole on one side. Then cut a second piece exactly the same size.
Place one piece of tape over the hole in the screen and press firmly. Depending on whether the screening material is metal or fiberglass, the tape won't stick very well, but that's okay! We're not done yet.
Flip over the screen and apply the second piece of tape to the opposite side. Now press the two adhesive faces against each other at the location of the hole.
Pardon my craptastic GiMP illustration: the black lines representthe screen with a hole in it (the gap) and the blue lines represent the two pieces of tape "meeting" at the site of the hole.
The tape doesn't have to
stick to the screen, just to itself.