Question: I currently have 14/180/65 installed, if I were to install a 15 inch rim with the factory specified tire size for the 15 inch rim would it raise my car height ? I also have 30mm lowering springs installed would it cause problems?

Answer: 
When you increase the rim size, you get a lower profile tire so that the overall outside diameter doesn't change. At the tire store and some websites this is known as your car's +1 (or +2, +3) tire size. 

If you don't change the tire size you'll run into several issues from minor annoyances like the speedometer being wrong to major issues like rubbing (and subsequent blow-outs).
Building off of JPhi's answer, a few tire options.

180/55R15 - same width, 1.7% smaller diameter  

180/60R15 - same width, 1.3% larger diameter  

185/55R15 - wider, 0.9% smaller diameter  

190/55R15 - wider, exact same diameter  

<https://tiresize.com/calculator/>

Some of these sizes may be more common than others. All of these sizes exist theoretically, but if it is not a common size, it is unlikely you will be able to find it anywhere, or you will have limited options.