No, it is not acceptable, even in colloquial or slang usage, to use the progressive form "I'm passing". To pass, pass on, pass up, or take a pass are all acceptable, but they are all simple tenses, because it is a complete action that one is performing. To pass is to say "no" to a specific offer or opportunity.
If, instead of turning down one specific offer, one made a decision to always turn down that kind of option, one would use a different word than "pass" to describe that. You might say one is "giving up" or "quitting" or "cutting out" or (somewhat more formally) "abstaining from" something like drinking alcohol if you mean it to be something ongoing (i.e. progressive or continuous).
Of course, in the broadest sense, people are likely to understand what you mean if you say "No thanks, I'm passing today", just like they understand all kinds of grammatically incorrect statements. Pretty much anything can follow "no thanks" and people will still understand the "no thanks". However, if you respond to "How about lunch?" with "I'm passing today" there is a fair chance you will not be understood.