Do you know the twist rate of the barrel on your rifle?
Because I believe the military .30-06 M2 is a heavier grain bullet. If I recall correctly, I think 1:10 was the prefered twist rate for the best accuracy, but that was also with a heavier barrel.
I am sure the Remington Long Action will handle the pressure if your weapon is in good condition and hasn't been abused. Having said that, the Remmy long action is a weaker action that a lot of the other magnum actions out there.
My honest suggestion, to completely error on the side of safety, is to have the weapon looked over by a competent gunsmith and have him give you the Go/No Go.
On the surface, the answer to your question is you should be fine, but there are variables that I would investigate before I spent a bunch of money on ammo for it.