Do I need to replace 10 speed chain if using a bigger cluster, also 10 speed? Most similar questions are asking about using the same chain with a bigger cluster with more gears (10 speed chain vs 11 speed cluster).

My concern is being able to simply buy a higher gear ratio cassette and still keep the same chain?

I have Shimano Ultegra 10 speed system, but my cassette is an 11-24 cluster. I want to replace it with another 10 speed cassette but 12 to 32, with grannies for massive hill climbs.

Surely I don't need to replace the chain if the number of gears on the cassette is the same - ie: 10 speed?
It all depends on how long your chain is currently. Also it depends on if your rear derailleur is a short or mid cage. I would say that a jump up from 24 to 32 is a bit to big for any chain. Of course without seeing how much slack is in your chain right now the right answer will vary. You certainly can put the new cassette on and you will know right away if it work or not. Be warned that you ABSOLUTELY do NOT want to add any more links to your used chain. This is a great resource for drivetrains in general: http://sheldonbrown.com/derailer-adjustment.html#chain