I have always found that when I switch from bench to offhand, I have to come up about 8 clicks or so depending on the rifle, as I've heard it explained; when the rifle is fired it buckels into a curve, and the barrel tends to point downward just a bit. When you are shooting from the bench with a front rest, the rest prevents the barrel from dipping down and therefore it shoots higher. but then when you shoot offhand the barrel is allowed to dip a bit more and therefore shoots lower. Harry Pope wrote a paper on this subject.

Bob

I find an awful lot depends on the rifle configuration and my rest technique.  If I am shooting an offhand rifle from the bench to sight it in, I try to put the front rest as close to where my left hand (I shoot right handed) would be as possible.  That way there is minimum change due to barrel deflection, change of barrel harmonics, etc.  As with all things though, YMMV!   

Froggie

As a general rule my rifles need 2 moa more elevation offhand than bench.  Less rim fire more high power.  I believe rifles recoil away from hard.

Boats

My own experience  matches what Boats wrote:

About 2 MOA, actually about the 20 ring.  The 22 RF little to none.  I've alway understood it was about the rifle shooting away (high) from a hard rest.  Same as we learned as kids not to rest the rifle directly on to of the fence post (stump, rock, etc) so not to shoot over the squirrel.  (To say nothing of scratching up the bottom of the stock!)

Over the years, when we have new shooters on the line, I've seen them stand up from the bench and shoot the offhand matches without making a scope change, and they end up with all their shots on the bottom half of the target. Many times for the better shooters, the target would have scored very well - if centered 4 inches higher! (200 yards)