Are there major obstacles to completing a thesis Masters program if I'm not funded/working in a lab? I will be entering an MS program in engineering this fall. I was not offered any assistantships or fellowships/funding. I was accepted to other graduate programs with some funding but chose to go to my top choice of school. 

I spoke to my academic advisor and they told me something very similar to 'most MS students here take out loans, complete a non-thesis program, and then go out into the real world and make a ton of money and pay them off.' 

Given that I would also like to apply for PhD programs and external funding upon completion of the MS program: what obstacles, if any, will be associated with me choosing to complete the thesis program if I will not be working in a lab? Could this cause issues when I go to apply for fellowships/PhD programs?
It strongly depends on your school, your department and your topic. Ask your advisor and/or PhD student what is the policy of the department.

There are theses where student work for a lab or factory and the thesis is part of this work (optimisation tasks, design of custom experimental apparatus), there are also, mostly theoretical, theses that can be succesfully defended and continued in PhD without strong bond with the department.

There are also topics, ususaly narrow-focused, where student is not skilled enough to work for the lab. Or there may be legal issues.