To properly answer this question you need to really give more information. You state the patient has Tb. Is this a strong clinical suspicion, PPD, smear positive, or culture positive diagnosis. Is continuing to test positive a continued positive PPD reaction, continued smear positivity, continued culture positivity or just clinical symptoms.
In general you can do three induced sputums on consecutive days either with hypertonic or normal saline or even just albuterol. These usually have to be done in some type of negative pressure environment because the assumption is you're aerosolizing a lot of AFB.
However, even having three negative sputum samples will not preclude a patient where your index of suspicion is high from undergoing bronchoscopy for lavage. And even if that is negative the patient if the index of suspicion is high enough will likely end up undergoing 4 drug therapy until the cultures come back negative.
Regarding the patient being contagious, if they're not coughing and coughing up a lot of AFB they may be infected but not infectious unless you're doing something to provoke them but even that is unlikely unless they have an abnormal CXR.
That's the "short" answer. It gets a bit more complicated too..