I have this friend I'm trying to get into asian dramas, but it's a little difficult to figure out what to show her because she doesn't like romance while a lot of the dramas I've seen included it. So I'm looking for some dramas that have either no romance or the romance isn't important that are engaging and interesting for someone who's never seen an asian drama to watch. I'm thinking of showing her Signal this weekend but does anyone have some other recommendations?

I think your idea of showing her Signal is the best choice. I also think Bad Guys would be a good choice. Maybe Mrs. Cop, there's some romance but it's subtle. If she's ok with medical dramas I'd suggest Romantic Doctor Teacher Kim or D-Day because the romance is there but it's not the main focus of the drama. 
I think Signal's a solid choice if your friend likes thrillers. I would also recommend Two Weeks, which has very minimal romance and a fast-paced plot. Wanted is another thriller drama with no romance, although I personally enjoyed this one less than Signal/Two Weeks. Special Affairs Team TEN is another option if your friend is a fan of American crime thriller/procedural shows.
Cruel City and City Hunter are also thrilling and engaging, although both of these do include some romance.
There is no romance in School 2013 or White Christmas, so if you are looking for a school drama, either of these might work.

My top pick is definitely Two Weeks. It has more of that addicting "k-drama" feel to it than Signal has, so I think it'd be a good first drama :) 
I'm still new to this whole drama scene myself, but the first drama that really got me hooked was the 2015 adaption of Death Note. The death note anime was already one of the best recommendations for anime beginners and the drama is even better, so I would suggest showing her that. There is also like no real romance in it.
38 Task Force is very good and doesn't have romance.
Signal, 38 Task Force and Pied Piper are good crime related shows that has no romance~
I Remember You has no romance, and is very suspenseful drama
I Remember You does have romance but it's a very minor aspect of the drama!

I recommend Misaeng!! It's a very good, realistic, slice-of-life drama with only the tiniest hints of romance
Like everyone else said, Signal. 
Also Misaeng because there's no romance in that but it is amazing