I've had the same problem in mIRC 6.12 resuming from a couple fserves (at least one of which was running Obsidian 0.9.2-pre1 on XChat).

Anyway I'm pretty new to mIRC scripting but wrote this small work-around last night. It's not perfect but I haven't encountered any problems with it yet.

To use this in mIRC, press Alt-R and copy the following block of code (between the lines of ===) into the Remote tab in the scripts editor.
Alternatively, download the script file from http://www.lethek.id.au/dccresumefix.mrc to your mIRC directory and type into your mIRC window: /load -rs dccresumefix.mrc

;==============================================================================
;I use quotation marks here since I don't know how spaces in filenames are handled or even if they're allowed by the protocol.
;Unless you know for sure that spaces cannot be used in the filename field of the message, I recommend you use this code.
ctcp *:DCC SEND:*: {
var %dccfilename = $ [ $+ 3- $+ [ $calc($0 - 3) ] ]
var %dccfullpath = $+(",$filename,")
if ($exists(%dccfullpath)) {
var %resumepos = $calc($file(%dccfullpath).size - 8192)
if (%resumepos < 0) { %resumepos = 0 }
.msg $nick DCC RESUME $+(",%dccfilename,") $ [ $+ [ $calc($0 - 1) ] ] %resumepos $+ 
}
}
;==============================================================================

;==============================================================================
;A shorter, faster version that doesn't bother about spaces. Untested- I don't know if it'll always work.
ctcp *:DCC SEND:*: {
if ($exists($filename)) {
var %resumepos = $calc($file($filename).size - 8192)
if (%resumepos < 0) { %resumepos = 0 }
.msg $nick DCC RESUME $3- $5 %resumepos $+ 
}
}
;==============================================================================

Raw debug logs using this script and resuming the download on a plain mIRC installation are as follows (with my address removed of course) -

DCC resume from mIRC 6.12 (sysreset 2.53) to mIRC 6.12 (plain):
<- :LethekServ!user@host NOTICE Lethek laughCC Send zx.cots.01.divx5.ogm (203.217.84.151)
<- :LethekServ!user@host PRIVMSG Lethek :DCC SEND zx.cots.01.divx5.ogm 3420017815 3287 238298721
-> irc.micrx.com PRIVMSG LethekServ :DCC RESUME "zx.cots.01.divx5.ogm" 3287 139976
-> irc.micrx.com PRIVMSG LethekServ :DCC RESUME "" 3287 139976
<- :LethekServ!user@host PRIVMSG Lethek :DCC ACCEPT "zx.cots.01.divx5.ogm" 3287 139976
<- :LethekServ!user@host PRIVMSG Lethek :DCC ACCEPT "zx.cots.01.divx5.ogm" 3287 139976

DCC resume from XChat 2.0.5 to mIRC 6.12 (plain):
<- :LethekServ!user@host PRIVMSG Lethek :DCC SEND zx.cots.01.divx5.ogm 3420017815 3530 238298721
-> irc.mircx.com PRIVMSG LethekServ :DCC RESUME "zx.cots.01.divx5.ogm" 3530 160984
-> irc.mircx.com PRIVMSG LethekServ :DCC RESUME "" 3530 160984
<- :LethekServ!user@host PRIVMSG Lethek :DCC ACCEPT zx.cots.01.divx5.ogm 3530 160984

I can't find a way to both stop mIRC sending it's own DCC RESUME message and to still receive the file. In any case there don't appear to be any adverse effects when the two DCC RESUME messages are sent, all clients I've tested with have no problems handling the two DCC RESUME messages.