Irelia is very strong laner and in my opinion is viable in any situation.

I run 9/21/0, and if u build boots first another option is going for 1/21/8 and getting MS quints. With that setup irelia will start off with 405 MS with boots.(I am still experimenting with these 2 options but i know for sure get the ms vs an easy lane)

armor pen reds
armor yellows
flat mr or mr/lvl blues(if u need mr in the laning phase get flat)
health regen quints

alternatives: health quints, MS quints, dodge yellows, armor/mr quints ect.

I get one lvl in each skill (q,w,e in that order most of the time) then max w, and afterwards max e.
sometimes u want e first for a lvl 1 fight or a lvl 2 gank or w first when u dont need q to last hit easier.
-sometimes i double rank w if i want more sustain and dont need the cc atm

I run ghost and teleport. The other option is flash instead of ghost or igite/exaust instead or teleport, and I like ghost better then flash for teamfights. The big question for Irelia is to go for ignite/exaust or teleport. If u think you might lose your lane get teleport and be more passive, and if you think u can kill your opponent more easily maybe run ignite or exaust(I like ignite better). For example vs renekton I would run teleport for sure but vs a gallio I would prob run ignite.

there are a few builds I do depending on their teamcomp and/or the person I am facing. For starting items I start with cloth 5, regrowth 1, or boots 3/4. I usually start with cloth though since top I usually need the armor and sustain in lane but vs ap i usually go boots and get regrowth vs an easy lane. The wriggles lantern in my build is situational on if need u need the armor and/or u need more sustain then your W gives. If u do not need wriggles get a philo stone and maybe start of with a regrowth pendant instead. When I get the easier lane I usually go a different build and get trinity force and 2gp10.
GA is also an option when using trinity force
thorrmail only sometimes when a ad carry is very strpng with armpen and is hard to focus(good tank line/excapes)

Edit: Since the mastery changes I don't like dodge boots or mercs for irelia very much. I would only get dodge boots if you need it for the laning phase or there are tons of auto attackers on there team. I would only get mercs if I need it badly for laning phase or they have a mass cc team (with irelia's passive and juggernaut the cc extra cc reduction is very little). Irelia scales with ats so i would almost always got beserker's greaves.

Edit 2: I added GA and last wisper to the situational items. For last wisper get this if the enemy team has a lot of armor and u need the GA or the Fon. I started using GA a lot for irelia's late game items and I love it, I would probably use it for a replacement for thornmail in most situations.

Edit 3: Removed Wriggles from build. Since the two nerfs to wriggles and health pots I don't like the cloth 5 start as much, and wriggles as an item. Irelia passive in almost all situations is enough sustain and I would only get wriggles if you don't need the magic resist or you need a lot of sustain to survive the laning phase.

laning as irelia is an art, some people fail hard at it and some succeed. There are so many things u can do in lane with her that I can not tell u because it would take too long, and these things u learn as you go. There are some simple things u can do as Irelia that I can tell that u need to know.
-knowing when to push and when to let them push
-when your lane starts to push early on when you have no ward vision b, get boots and pots/ward and tele back
-perfect last hitting at the tower with her since she has a lot of tricks to do(and csing in general) u can auto attack and q in about the same time which makes it very easy to last hit at the tower
-double gap closer using an injured creep(granted harder to do since I don't max q till last)
-learning when to be aggressive and when not to, like when ur ult is up trade with them unless u think u may get ganked
-never open with q in lane, always e them, w and q to them when they try to run away for a free hit or 2

This spell you need to use correctly and by this I mean sometimes it is good to use your teleport to go back to lane faster and other times you should save it for dragon control or joining teamfights and ganking (always use your first teleport to go back to lane and usually your second).

Irelia job in team fights is as a anti carry, and with this in mind you need to stick on to mainly their ad carry and melt them then immediately switch targets to the next squishy. Irelia is bad at killing tanks but she melts squishes really fast.
- don't be afraid to use ghost in team fights
-always take the red buff

edit: Just in case anyone is interested I smart cast everything on Irelia.