What minimal resistance should be between an input and output pin not to burn Arduino?
Looking towards the high end of recommended (vs absolute maximum) current and considering Ohm's law at a 5v difference if two outputs are trying to drive each other to opposite values, you would probably get away with as little as 250 ohms.  But in most cases that would needlessly waste power, and in the case of multiple instance you could start totaling towards the limit of power supply current for the chip overall.

Typical values for external pull up resistors are more in the range of 1K-10K though higher is not uncommon - the internal pullup which you can optionally activate by setting the output state of an input pin high has a value of several tens of kilo ohms.  Lower values will consume more power, higher values may be less reliable in a noisy situation.