mIRC checks for the ALT+Tab key combination and does not display the menubar here in that situation, under either Windows XP or Windows 7.

I just tested a debug version of mIRC under Wine. When you press the ALT key, mIRC detects the key press, as it should. However, the moment you press the Tab key, Wine captures that and all subsequent keypresses, even if you ALT+Tab back to mIRC and release the Tab key. All mIRC sees, at a keyboard hook level, is the ALT key being pressed and released.

So it does not look like there is a way for mIRC to work around this particular Wine issue.