If I remove the heatsink/fan...

You can still use the old one yeah   should not be a problem

it would if it was 2 different pastes, but if the heatsink is clean and there is paste on the CPU then it should work as normal  

Anytime you remove the heatsink it is highly recommend that you clean it and the CPU and reapply a fresh thin layer of thermal paste.

Removing the heatsink breaks the thin paste layer which helps improve heat transfer. It introduces air pockets and that acts as a heat insulator rather than a heat conductor.

^+1 You take it off, you clean and re-apply new paste
Moto

Clean it thoroughly and re-apply fresh paste do not re-use.
http://www.arcticsilver.com/methods.html

Thanks so much all!!

It should still work.
You should change the thermal paste\pad every few months, but if you already did in the past few months, you should be good.

Yes you can re-use pads but not paste, if it has a pad I'd remove it and use paste anyway.
You don't need to re-apply paste every few months just when needed, like your cpu temps rise.