A safe dolly is a must, along with some of the other equipment necessary.
And regular furniture movers might not be the best choice for safe moving. Call your local locksmith and ask if he does safe work. Some questions to ask:
1) What brands of safe are you familiar with?
If he says Mosler, Hayman, Diebold, American Security, or Lord, you're in business.
If he says Sentry....well, he wouldn't be my first choice.
2) What is the largest/heaviest safe you can move?
The correct response to this one is, "It depends greatly upon where you want to move it."
I'm trying to help you feel out the kind of pros who really KNOW what they're doing vs. the dolt who just wants to take your money and "learn by doing" using YOUR safe.
I chose a 3600 pound TLTR30 safe. That is not a typo. The safe really is heavier than some cars. I moved it myself with a Johnson Bar and a pallet jack. It's on my porch. I don't figure anyone's going to steal it anytime soon, and it's rated higher than any structure I could think to put it in, short of perhaps a bomb shelter or missile silo, or maybe a bank vault.
The advice given here is right on IMHO. The trick is not to get professionals, but to get SAFE PROFESSIONALS to move that safe. The part about removing the door is also correct, ESPECIALLY considering that many relockers are NOT resettable from the exterior without destructive drilling.
Good Luck,
Panz