Equal voltage, but not shorted?
For the question in the first paragraph

A clarification to my earlier statement. You have to be careful how you explain things. If you assume that "two nodes are always kept at the same voltage by some unknown force", then the problem is simply, no it will not have an effect - because as you stated, the two nodes are always kept at the same voltage no matter what - they must be independent of each other for this to be the case.

However, just because two nodes are the same voltage, doesn't mean that a wire will have no effect. In a circuit where two nodes are dependent on each other, then just shorting them out may or may not have no effect. Why? because the wire may change the transfer function of the circuit and thus the relationship between the two dependent nodes. This is certainly the case in op-amp circuits where you have feedback.

In regards to the second paragraph

Op-amp terminals aren't shorted internally (you can short them externally, but it's not particularly useful to do so). The voltage is not always the same at each terminal - for example you can connect one to one supply rail and the other to the other supply rail. However the effect of having them non-equal is that your op-amp output clamps to one or other of the supply rails. This is because op-amps typically have very high gains - a small difference in input voltage results in a large difference in output voltage.

I think you are confusing concepts. There is an approach with op-amps of negative feedback in which the output feeds back in to the negative input terminal. Any changes made to the input terminal will result in the output value trying to change, but because you have feedback, the output then affects the input which is typically designed so as to bring the input difference back to zero at which point the output stabilises at its new value.