Short stub transmission line

If I short a stub to ground on a 1.5mm FR4 board, they say to connect a line from top to bottom, but wouldn't that change the characteristic impedance and hence reflection would increase?

> wouldn't that change the characteristic impedance ...
Yes!

> and hence reflection would increase?
Well, that depends on what your stub is doing.

Concentrating on the first part then, yes, you need to model the effect of the via(s) linking the upper microstrip layer to the underlying ground plane - these become part of the overall equivalent circuit of your 'stub'. Most (all?) 2.5/3D EM solvers can model the via geometry when simulating your layout. Many packages (ADS, for example) also have comprehensive 'schematic' models that will estimate the parasitics associated with the shorting vias and help you adjust your design without the computational overhead of a full 3D EM simulation.

...and if you're just tinkering with microwave circuits with no tools/interest/need for comprehensive simulations, then simply use as many vias in parallel at the end of your stub as your geometry/density permits and hope for the best!

(Paralleling the vias serves to reduce the associated parasitic inductance and resistance of the short to closer approximate the desired ideal short behaviour).
(Afterthought #2 : General purpose FR-4 is a notoriously difficult board to accurately model in an EM simulation if you haven't got tight reins on your PCB manufacturer/substrate parameters. If you're going to go to the effort of doing comprehensive simulations, use a better substrate :)

if you have a stub transmission line, and short circuit the far end...yes the reflection coefficient will change. It will change to a magnitude of 1 for that stub.