I was watching a video exercise to play all of the modes you can over a power chord A5 in this case. Is there an interval that all modes share over a particular tonic (A in this example)?
To answer this, we can arrange the modes in order from those that have the highest-pitched notes (largest intervals relative to tonic), to those that have the lowest-pitched notes (smallest intervals relative to tonic), then compare the resulting intervals. Note how, in this order, each following mode is identical to the previous one, except for one scale degree being a half-step lower.
In the scale patterns, I've bolded the half-steps (H), so you can see the pattern of how they shift left each time. In the intervals, I've bolded the one that is different from the preceding scale.
- WWWHWWH = M2, M3, A4, P5, M6, M7 (Lydian)
- WWHWWWH = M2, M3, P4, P5, M6, M7 (Ionian)
- WWHWWHW = M2, M3, P4, P5, M6, m7 (Mixolydian)
- WHWWWHW = M2, m3, P4, P5, M6, m7 (Dorian)
- WHWWHWW = M2, m3, P4, P5, m6, m7 (Aeolian)
- HWWWHWW = m2, m3, P4, P5, m6, m7 (Phrygian)
- HWWHWWW = m2, m3, P4, d5, m6, m7 (Locrian)
As you can see, each interval changes at some point, so there is no common interval (other than the trivial unison/octave). However, you can also see which ones are most stable. Specifically, the perfect fifth occurs in every mode except Locrian. Some musicians don't even consider Locrian a true mode due to it's lack of a perfect fifth. If you leave it out, then all the remaining modes have a P5. Similarly, all modes except for Lydian have a P4. If you again throw out Locrian, then Phrygian is the only other mode that has m2; all the other modes have a M2.
A nice addendum to Caleb Hines' answer is that if you take all the most common intervals, you get M2, m3, P4, P5, M6, and m7, which is the Dorian mode. What's significant about this is that the Dorian mode is a point of symmetry in our diatonic scale. If you use D as a center point and move both up and down in perfect 5ths, you end up getting the diatonic scale after 3 goes. So D can be a literal tonal center and any interval present in one direction will also be present in the other.