A Lot. Like A Lot, A Lot.

And many who otherwise would have mustered out or signed back up were Dead or Invalid after the Overland Campaign. From early May 1864 the Army of the Potomac and attached 9th Corps numbered about 120k, by late June as they settled into works around Petersburg it was at 86k and in the middle had suffered over 60k casualties of one variety or another, and stripped the garrisons of DC and other points like New York and Boston. Other actions included recalling a large part of 19th Corps from the failed Red River Campaign to Virginia, additional recruitment of USCT, and the continued threat of the Draft, and liberal leave and a large cash payment for those veteran units who elected to serve out the war. Bounty Jumpers though were a mjaor problem, and while desertion rates for soldiers with actual time in service had been pretty well addressed under Hooker the year before(better conditions in camp, leave for the men, etc) it could still mean that a regiment might lose 2-30% of any batch of recruits!

Also worth noting that some units suffered much worse than others, 2nd Corps in particular found itself the tip of the spear on several occasions, one of its divisions under John Gibbons suffered a 70% casualty rate among the men he took with him into The Wilderness to begin the campaign. The knowledge that many of these regiments were coming done also contributed to the momentum, among pressure to follow up Gettysburg, to the aborted Mine Run campaign in late 1863. Hooker had also been faced with a similar dilemma with shorter service units the year before.

But as to your direct question on the 3 Year Regiments, yes the AotP hemorrhaged many of them throughout the Summer of 1864. June was a big month with 30 regiments who had not resigned due to depart as an easy example I have on hand! Even if each was much reduced at say 300 men each that is 9000 men, an extremely large division or two small ones right there.

The good news is that even before the campaign began about 30k had resigned, and in all about half did. But that still meant the hard core experienced part of the Army of the Potomac in Fall 1864 was maybe 1/3 what it had been in April, the rest were pulled from other sources or newly recruited. And not just men but officer attrition had been extremely high. By the time Petersburg arrived even the men not about to depart were worn out, demoralized in many cases, and fighting under unfamiliar officers possibly alongside unfamiliar faces atop the missing. And as the men wised up to the newly fashionable pattern of attempts at a flanking march, and then confronting hastily or not so hastily dug in rebels and possibly attack the same point 2-3 times attacks began to be made less enthusiastically in many instances(though many were still carried with extreme force like Hancocks assault at Spotsylvania).

I would ping /u/DanDan_noodles too, absolutely a top notch guy in this area.

An excellent read on all things AotP even though it is more group biography is Stephen Sears Lincoln's Lieutenants!