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To use winecfg to automatically enable/disable dbhelp.dll:
1) Run winecfg. (If you have multiple prefixes, make sure it is the correct prefix.)
2) In the 'Libraries' tab type dbghelp into the 'New override for library' box.
3) Click 'Add', then 'Yes'.
4) Click on 'dbghelp' in the 'Existing_overrides' list.
5) Click 'Edit'.
6) Set to 'disabled'.
7) Click 'OK'
(You have now disabled dbg help to get the Battle.net Client to work. Now we need to re-enable it for 'Diablo III.exe'.)
8) Go to the 'Applications' tab.
9) Click 'Add Application'.
10) Select your 'Diablo III.exe' and click 'Open'.
11) Make sure 'Diablo III.exe' is selected in the list.
12) Switch to the 'Libraries' tab.
13) Type dbghelp into the 'New override for library' box.
14) Click 'Add', then 'Yes'.
15) Click on 'dbghelp' in the 'Existing_overrides' list.
16) Click 'Edit'.
17) Set to 'builtin'.
18) Click 'OK', then 'OK'.
(You have now re-enabled dbghelp.dll for 'Diablo III.exe' only)

Note: The above steps will disable dbghelp for _all_ programs other than Diablo III. It is suggested that you use a separate prefix for Diablo III if you are going to do this.
It is hard to tell what application(s) need dbghelp disabled, so it is easier to just enable it for 'Diablo III.exe'.

Edit: If you are not, you should really be checking/updating WineHQ ... I've had this solution there for days.