Does the drainage of a ovarian cyst cause infertility?

Unlikely. Ovaries always produce cysts, and some of them for a reason or another enlarge and cause trouble
Drainage of cyst does not cause trouble UNLESS the whole ovaries are taken out or no much tissue is left in them due to the original disease.
No. I wish these were all this easy to answer. A cyst big enough to be worth draining is going to be trouble for you; at the very least, it'll be uncomfortable and there's the risk of it twisting and sending you to emergency surgery.