The confusion comes from the fact that 'after noon' and 'afternoon', while looking looking similar, do not refer to exactly the same thing.
For example, 11PM is technically 'after noon' - noon has passed. But it isn't within the time period considered to be 'afternoon' (generally ~12:00PM-5:00PM).
I'm not here after noon
That states that you won't be around once the indicated time has passed. Not in the evening, or at night.
I'm not here after the noon
After 'the' noon sounds odd, because there is only one noon every day - no need to specify which noon we're talking about.
I'm not here afternoon
This doesn't make any sense. You should say "I'm not here in the afternoon", because you're talking about something within a designated time period. Depending on the schedule you keep, this leaves the option open for you to be here in the evening or at night.