Can the saturation of an OPA influence its input?
It certainly can - depending on the opamp.

An opamp in linear operation maintains the same voltage on both inputs. With a bipolar input stage, that leaves Vbe on both input transistors the same.

Outside the linear range that is no longer the case. 

Consider the case where saturating the output leaves Vbe too low on the feedback input. It follows that Vbe on the signal input can be too high, leaving that base-emitter junction conducting heavily, acting as a forward biased diode, where it should be a high impedance input.

In your case, feed the third opamp from the measurement point via a high value resistor (maybe 10k) : this will both buffer the measurement point from any such effect, and (by measuring both ends of the resistor) indicate whether this problem is actually occurring, with minimal effect on the signal seen by the third opamp. (Check R and the opamp's Cin meet your bandwidth requirements, of course)