PICAXE very short interval timer (for playing sound)
If you want to do something regularly every 23 &micro;s, use a interrupt.  You didn't say what PIC your board uses or what its clock rate is, but even if it's a old 16F at 20 MHz clock, it can still do 5 MIPS, or 5 instructions per microcsecond.  That would leave 115 instructions per 23 &micro;s interrupt.  That could be tight if it has to go to the EEPROM every time.

If this algorithms only needs a individual bit every 23 &micro;s, then perhaps a foreground loop can read the EEPROM a byte at a time into a FIFO and the interrupt routine only process a single bit each time with one FIFO fetch every 8 interrupts.  That should be quite doable in well under 100 instructions.