I've had success with Sysinternals Process Explorer. With this, you can search to find what process(es) have a file open, and you can use it to close the handle(s) if you want. Of course, it is safer to close the whole process. Exercise caution and judgement.
To find a specific file, use the menu option Find->Find Handle or DLL... Type in part of the path to the file. The list of processes will appear below.
If you prefer command line, Sysinternals suite includes command line tool Handle, that lists open handles.
Examples
c:\Program Files\SysinternalsSuite>handle.exe |findstr /i "e:\" (finds all files opened from drive e:\"c:\Program Files\SysinternalsSuite>handle.exe |findstr /i "file-or-path-in-question"
Well, we finally appear to have resolved this issue in our environment. For the benefits of others, here's what we discovered and how we fixed the problem:
To try and gain further insight into what was occurring before/during/after the delays we used Wireshark on a client machine to capture/analyse network traffic whilst that client attempted to access a DFS share.
These captures showed something strange: whenever the delay occurred, in between the DFS request being sent from the client to a DC, and the referral to a DFS root server coming back from the DC to the client, the DC was sending out several broadcast name lookups to the network.
Firstly, the DC would broadcast a NetBIOS lookup for DOMAIN (where DOMAIN is our pre-Windows 2000 Active Directory domain name). A few seconds later, it would broadcast a LLMNR lookup for DOMAIN. This would be followed by yet another broadcast NetBios lookup for DOMAIN. After these three lookups had been broadcast (and I assume timed out) the DC would finally respond to the client with a (correct) referral to a DFS root server.
These broadcast name lookups for DOMAIN were only being sent when the long delay opening a DFS share occurred, and we could clearly see from the Wireshark capture that the DC wasn't returning a referral to a DFS root server until all three lookups been sent (and ~7 seconds passed). So, these broadcast name lookups were pretty obviously the cause of our delays.
Now that we knew what the problem was, we started trying to figure out why these broadcast name lookups were occurring. After a bit more Googling and some trial-and-error, we found our answer: we hadn't set the DfsDnsConfig registry key on our domain controllers to 1, as is required when using DFS in a DNS-only environment.
When we originally setup DFS in our enviroment we did read the various articles about how to configure DFS for a DNS-only environment (e.g. Microsoft KB244380 and others) and were aware of this registry key, but had misintepreted the instructions on when/how to use it.
KB244380 says:
The DFSDnsConfig registry key must be
added to each server that will
participate in the DFS namespace for
all computers to understand fully
qualified names.
We thought this meant that the registry key has to be set on the DFS namespace servers only, not realising that it was also required on the domain controllers. After we set DfsDnsConfig to 1 on our domain controllers (and restarted the "DFS Namespace" service), the problem vanished.
Obviously we're happy with this outcome, but I would add that I'm still not 100% convinced that this is our only problem - I wonder if adding DfsDnsConfig=1 to our DCs has only worked around the problem, rather than solving it. I can't figure out why the DCs would be trying to lookup DOMAIN (the domain name itself, rather than a server in the domain) during the DFS referral process, even in a non-DNS-only environment, and I also know I haven't set DfsDnsConfig=1 on domain controllers in other (admittedly much smaller / simpler) DNS-only environments and haven't had the same issue. Still, we've solved our problem so we are happy.
I hope this is helpful to the others who are experiencing a similar issue - and thanks again to those that offered suggestions along the way.
Best Answer
The closest you'll be able to come is with a DFS namespace, which you can set up to ensure you don't have to update all your UNC links next time you change which server(s) host your shared files and folders.
As you noted, you won't be able to use the same path you use now, which is the server's hostname, but you will be able to have a namespace you can manage and transparently move hosts in and out of as you need.
Otherwise, as noted in MichelZ's answer, it is not possible to use DNS to direct different services to different hostnames. DNS is just not set up that way - it registers hostnames to IP addresses, and is unable to selectively translate the same hostname to different IPs.