[Edit: this question has pretty much been answered but I figured I would share the answer I drafted yesterday anyway.]
It might be easier to think about it in terms of image quality since DPI is mostly applicable to print (the 'D' being dots of ink); to answer your question, Inkscape will not discard picture information (essentially number of pixels) unless you tell it to (although it should be noted it renders PNGs somewhat poorly live, it does it better when you go to rasterize when you export [if you do, rather than just saving as a vector format with PNG embedded as Base64]). If you resize it, there will still be the same number of pixels; therefore the DPI will increase as the total absolute image area decreases (in a parabolic manner, because DPI is a square measure (dots per square inch).
I think there was another point I was going to make but I forgot what it was.
-Arlo James Barnes