I have Shimano Ultegra 6600 rear derailleur with a 10 speed 12-25t cassette which I am looking to change for a 10 speed, 11-34t or 11-36t. Can someone tell me if this will work without changing the derailleur?
No, the pull ratio on the 11 speed derailleurs is different so it won't work, even if you ignore the extra gear. You can probably pick up a GS cage 10 speed 105 or Ultegra derailleur on ebay pretty cheaply.
Other options are:
11 speed shifter, cassette and derailleur (you don't need to change the crankset, front derailleur or shifter, they'll be fine).
Smaller chainrings (6600 is 5-bolt though so this will likely mean a new chainset).