Can dulera (formoterol and mometasone) lose it's effectiveness over time? I'm on dulera (formoterol and mometasone) 100mcg 2 puffs 2 x a day and just got a cold and need to use rescue inhaler throughout day

Not the medication. Something (eg. Viral infection, post nasal drip, smog, acid reflux) is triggering your asthma. Its presence will continue to trigger the attack. The medication will control each episode but unless the trigger is removed, your symptoms will continue or worsen; see your doctor to help determine if your meds need to be adjusted or another med started.
Possibly. However I would lean toward it's not working well because Dulera doesn't work well. I never use that drug for anyone.
Virus. Virus infection is known to be a trigger for asthma attacks and also strongly suspected to be a major causative factor in asthma. Thus during a viral infection, you usual medication is often insufficient and many do require a short course of cortisone (prednisone) to reduce the obstruction and restore the airway's response to rescue medication. Contact your doc asap.
May need more. Talk to your doc. You may just need more rescue to to illness. If need for inhaler that high your doc may want you on oral steroids.