So you are saying that the TinyISP sets the ISP pins to inputs after uploading the code to the ATtiny
Yes. RESET is released after the SPI pins are reconfigured as INPUTs. The target can only run if the SPI pins are INPUTs. The code of interest is in end_pmode
After uploading, it may be possible for the internal pull-up to be enabled on SCK or MOSI but that would indicate a problem with the SPI circuitry (very unlikely). If the SPI circuitry includes separate output drivers then that situation is impossible. With the exception of a loose wire on RESET and a stroke of very bad luck, I cannot think of any failures that would allow the target to run with the SPI pins configured as OUTPUTs.
If you're still concerned, put a series resistor (250 ohms or more) on MOSI and SCK. In the event of a problem, the resistors will protect all the players from damage.
and that any connections on the ATtiny using those same pins (I2C) to a real-time clock should be able to function?
You may have to isolate the RTC during programming. I have to idea how an I2
C device reacts to "random" activity on the bus.
Does the RTC have a RESET? Or an ENABLE? You may be able to use that to keep the RTC quiet during programming.
In any case, there should be no need to disconnect / reconnect the programmer.