One of the problems with donations is they need to be processed, i.e. allocated, jacketed, catalogued, spine labelled, added to computer system, security label added, etc, etc. It doesn't sound a lot but it all adds up in man hours. Where I worked (one of the London boroughs) we've been lucky enough not to have any libraries closed, in fact the hours have been increased slightly. BUT staffing has not been increased (having been reduced previously over the years), just spread more thinly to cover the hours open. The qualified staff at headquarters have been drastically reduced. We simply did not have the staff available to deal with donations.
When we did previously take donations often the books we were given were in no fit state to be added to stock, or were older non-fiction editions. I once accepted about 50 foreign fiction which I was assured had only been read once and were in excellent condition. Only 2 were suitable to be added to stock. To use Terribull's description, often the unwanted tatty books were included as donations because charity shops wouldn't take them, folk didn't want the responsibility of chucking them away themselves so passed that onto library staff by 'hiding' them in the bags of donated books.
Personally I believe libraries should take donations. But how do you prevent ending up with multiple copies of the best selling authors? These are the titles that libraries tend to buy anyway.
I won't go on as it was something I was passionate about but had little or no control over. All I can suggest is get in touch with the acquisitions (or equivalent) librarian at the headquarters of your county/borough and ask if they have a Donations Policy, and would they consider implementing one if they didn't.