For hypothyroid patients on dual replacement therapy (levothyroxine & liothyronine), do you recommend monitoring TSH while holding off on the morning T3 dose?  

In my experience, there are a very few indications for using a combination of T4 and T3. However, if that is the case there is no need to stop T3 prior to getting a TSH level. TSH will reflect the metabolic status of the patient.

No, I do not withhold the morning dose of T3. Older studies have shown no effect of an oral dose of T3 on serum TSH levels (Saberi & Utiger, PMID 4422006).

I frequently will offer a trial of combination therapy to patients with clear symptoms despite a normal TSH on an appropriate dose of LT4. I do not specify whether they should take their morning T3 or not before labs - while data does show it probably makes a difference, the difference is small.