The answer is an unequivocal "maybe." It depends on why the tapes fell off in the first place.
For example, if they fell off because of physical abuse, or they weren't attached too well to begin with (separate issues), then they probably won't affect the recycled solution.
If they fell off because of adhesive failure caused by the wash solution, then absolutely they will contaminate the recycled cleaning solutions. A wash solution (whether aqueous or solvent based) may attack an adhesive bond, in one of several ways.
Typically, aqueous wash solutions contain small amounts of detergents or co-solvents, to lower the surface tension of the cleaning solution. This allows for better "wetting"of the various surfaces on a subassembly, which is necessary to provide intimate contact of the cleaning components with these surfaces.
If the cleaning solution attacks the adhesive bond, it is usually due to either,
a. absorption of the detergent or solvent by the polymer materials in the adhesive, causing them to soften and/or swell (which weakens the adhesive bond); or,
b. it either extracts/dissolves components from the adhesive into the solvent, which not only weakens the adhesive bond, but also results in soluble materials in the cleaning solvent.
In the former case, there may be little, if any contamination of the solvent by leaving the tape in the solution. However, in the second instance, problems will occur. In this instance, repeated use of this same solvent, will increase the concentration of the soluble materials, if the adhesive actually dissolves in the solvent. This will subsequently increase the amount of residue left on the products, from the cleaning solutions!!
Usually adhesive attack occurs only at the edges of the adhesive bond, in contact with the solvents. By leaving the tapes in the solvent, the entire surface of the tape is now available for the solvent effects to occur more thoroughly.
Once something dissolves into a solution, it stays there, unless the solvent in that solution can be purified, say by distillation, or ultra-filtration/reverse osmosis processes. So, if something is dissolving, as each new board comes through, the concentration of the dissolved material increases.
If the tape falls off because of these latter defects..CHANGE TAPE! In any case, I'd suggest removing the pieces from the baths.