Depends on what kind of drip system you mean. You can absolute do it afterward, since mulch is easy to move, but you should pile the mulch over the irrigation again afterward, to protect it from the sun (just be careful not to accidentally cut it with a shovel in later years =P).
If you are going to mulch the entire yard, bear in mind two things:
A) It won't stop weeds unless you mulch very deep (multiple inches at least), they'll just work their way through.
B) It'll eventually let weeds through anyway.
I'd save the mulch for the areas the plants are actually going to grow, since it'll improve the soil over time, and not waste it on the places you don't want anything to grow.
Last year what I did was laid down cardboard first, and then woodchipped ontop of it about an inch. That kept weeds down perfectly for a year. This year the weeds came back as if last year's labor wasn't a thing.
This is excellent if you are going to plant there next year (or even this year - just cut a hole in the cardboard).
So, this year for pathways I laid down cardboard first and two layers of Agfabric landscape ground cover on top (that particular link is expensive, because it's 6.5' wide and 300' long, but it comes in other sizes).
I can't personally vouch for the Agfabric as this is my first year using it, but my local greenhouse uses it and it seems to last long and avoids alot of the issues other forms of weedblock fabric have (which deteriorate and get tangled up in weed roots several years later). As mentioned, I doubled it up just incase, with cardboard under it. =P
I buried (just 4" deep) several runs of irrigation lines under it, running extra lines to prep for future expansion.
If you haven't already done so, now would be a good time to lay out your garden on paper (or on the computer), and figure out what you want to do this year and future years (or general guesses of what you might want in the future), and plan your irrigation system in accordance with that.