Can any damage be done by
over-enthusiastic use of WRs?
Yes it can.
One concern is that weak references make your code more complicated and potentially error prone. Any code that uses a weak reference needs to deal with the possibility that the reference has been broken each time it uses it. If you over-use weak references you end up writing lots of extra code. (You can mitigate this by hiding each weak reference behind a method that takes care of the checking, and re-creates the discarded object on demand. But this may not necessarily be as simple as that; e.g. if the re-creation process involves network access, you need to cope with the possibility of re-creation failure.)
A second concern is that there are runtime overheads with using weak references. The obvious costs are those of creating weak references and calling get
on them. A less obvious cost is that significant extra work needs to be done each time the GC runs.
A final concern is that if you use a weak references for something that your application is highly likely to need in the future, you may incur the cost of repeatedly recreating it. If this cost is high (in terms of CPU time, IO bandwidth, network traffic, whatever) your application may perform badly as a result. You may be better off giving the JVM more memory and not using weak references at all.
Off course, this does not mean you should avoid using weak references entirely. Just that you need to think carefully. And probably you should first run a memory profiler on your application to figure out where your memory usage problems stem from.