On a Remington 700, you'd have to change the bolt, or have the bolt face opened up to accomodate the .532 rim diameter of the .375 H&H. The cartridge is longer than the .30-06 and I don't know how much room you have in that long action. A barrel swap, for sure. The limiting factor would be the action length. I'm just not sure if a Rem 700 action has enough room in it for the longer cartridge.
However, I"ve seen a .416 Rigby built on a Savage long action. Savage makes their long actions long enough for magnum cartridges. The bolt face is a standard magnum bolt face and Savages are designed for the barrel swap. Any donor action from any local pawn shop could be used, and used Savages sell at the low end of the used rifle market.