Will it hurt to replace the 20 amp fuse in my cigarette liter to a 25 amp it keeps blowing fuses

If it keeps blowing fuses, you have a problem. The fuses are there to protect the wiring. If you put a larger fuse and it doesn't blow, you might overheat something, or worst your car could catch on fire.

I am wondering if the fuse blows instantly or whether it lasts a while?

Fuse ratings for cars are usually mean or average values so a 20 amp fuse needs roughly 40 amps to cause it to instantly fail. I have not heard of a cigarette lighter needing that much current so I guess there is a fault that should be located and cured before you have a fire.
Maybe it is the element that is faulty...

I think fitting a 25 amp fuse would be slightly foolish...