Speaking just from my experience with people in DOD and DOJ, a lot do (Everyone I know certainly has), at least in the professions as opposed to trades, though technically it violates some laws including the Anti-Deficiency Act (The set of laws which prohibit agencies from spending more than Congress has appropriated and authorized for them), since that unfunded work creates an obligation on the agency's part to pay for the value of the services received, but it hasn't budgeted to pay for that. Since it is the employee's regular job, and longer-term issues like ratings and promotion might be affected, any such unpaid time can't be regarded as a truly voluntary 'Gift.' As far as the gift issue itself goes, Congress severely limits the ability of agencies to accept gifts that further their mission, since Congress wants to be able to control what they do via the 'Power of the purse,' so Congress regards any such gifts as 'Unlawful supplementation of an appropriation' issues beyond a certain very low threshold ($1000 in the Army, for appropriated fund activities; nonappropriated fund activities benefitting morale, welfare, and recreation instead of the actual agency mission have a much higher acceptance authority).
In a lot of ways being a Federal professional (Doctor, lawyer, comptroller, engineer etc.) is like working a job with the worst features of hourly employment and the worst features of salaried employment combined...all the pay and benefits are based on hours only, you have to account for every hour each two-week pay period, your leave is based on hours, there is almost never any overtime authorized and it actually pays less than straight time for a lot of the long-service pros, there is no meaningful performance reward in the sense that it exists in the private sector (And in a way just as well, because since there is not a meaningful way to judge ROI for government services, it would devolve into an office politics game pretty quickly, which is what has happened with the rather modest incentives under NSPS in many places), but the demand or mission is still there and has to be met whether the resources are sufficient or not.
It is by the way a myth that it is impossible to fire a Federal employee, it is probably easier than firing an employee in a private sector union shop. But, like managers in a union shop, Federal managers by and large don't see it as worth the battle to take a disciplinary or performance action short of firing, so they wimp out on doing any of the required preliminary work building a record of problems until it becomes a crisis. Then they piss themselves whining about how 'Unresponsive' the system is to their 'Needs' when they can't go straight from 'Unblemished record' to 'Fired' in one leap on the problem child.
