What does measuring voltage do to the current or signal of a pin?
A voltmeter acts like a high-value resistor and a small-value capacitor between the two points where the probes are attached. If one of the probes is on ground, then you could consider this a pull-down resistor. The resistor will be in the 1M\$\Omega\$ range for an inexpensive meter, much higher for a good lab-quality instrument. If the point where you place the probe is floating then the resistance can be significant. It usually doesn't matter for logic circuits.

The capacitance may have an effect on high speed logic signals. I don't know, but I would guess that the capacitance might be on the order of 100pF or so.

A third effect, that wasn't on your list, is pressure. When you press the probes onto a circuit node you may improve a poor connection. This is common when there is a poor solder joint, and the pressure of the probe forces the connection to be made.