If I have 3 transformers (120V transformers) in series (on the input side) on a 240V circuit, will the transformers experience a voltage drop? If so, how much and will this affect the secondaries by the transformer equation?
Nope, this is under-constrained. You have not said the the transformers are identical and, because of this, the big problem is that putting the primaries in series does not mean equal distribution of voltage to each primary.
This immediately means one transformer might expect more core saturation than another and overheats and fails. And, all of this is before you analyse any effect due to secondary currents. If secondary currents are unequal then you have an even worsening scenario and you might expect a fire.
Don't do it - test the transformers first and ensure they are VA and primary magnetization current compatible. Then ensure that any secondary current is identical for each.
Even if you obtained three transformers that were of the same make and model, the primary magnetization inductance might vary quite a bit between them and this could cause one of them to exceed the 120V AC rating even though, on the face of it, you might expect all three to have 80V AC across the primaries.