Ok, so we have all read on here that to build speed you must practice slow, with a metronome, and try to have economy of motion. Well here is my story. Keeping it short

My guitar teacher at school (Im in college btw) tells me to try to push my speed, even if i miss notes, and dont get them all. Either on songs, scales, melodies etc etc. To try to get them use to going faster.

Usually when i practice ill practice it slow and build it up get it on time with a metronome. Then do speed bursts. Example play a melody slow 4 times, then play it faster 2 times (making sure its comfortable and clean), and the back to slow. Which has helped me play faster. I know my speed will come, and i know there isnt any short cuts to becoming super fast.

My question is should i take this advice? i know im suppose to listen to my teacher, which i do! But im just confused on this...
How far should i push it?

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Push your pace, but not from something like 120 BPM straight into spastic arm movement at 200BPM.

Play your licks normally, as comfortably as possible at a specific tempo. Then, after playing the lick multiple times, increase tempo marginally. This means by less than 5. This is a small hurdle to adjust to, and you should become comfortable with it shortly.

Its much better to bring speed up slowly and well than quick and sloppily.

If the licks/scales/Whatever are alternate picked, make sure your being economical and using almost entirely wrist.
i find sometimes that playing at the limits of your techiue just out of your comfortable zone is helpful to see whenre you have mistakes in your technique that you could fix
also is a humbling experience
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Its much better to bring speed up slowly and well than quick and sloppily.

^ I agree with this. in my opinion there is really not a whole lot of benefit to practicing faster than you can control.
That does pretty much nothing for your actual development toward becoming a fast player (if you want to sound good and clean).
Playing out of control is the opposite of what successful fast players do.

I wouldn't say your teacher is wrong... there's some well respected authors who advocate over-playing your ability, then slowing back down so it seems easier. I guess it can't really hurt you once in a while.
I just don't buy into it personally.
Neither does Steve Vai and some other famous players... (straight from their own words)

And when you do practice slowly... think about what will make that piece harder at higher speeds. Like sliding position changes... make them fast and razor sharp even at slow tempo... remember it's the tempo that is slow - not you.

The whole purpose of slowing down is to develop perfect execution for higher speeds. So concentrate and practice perfect, precise execution at slow tempos... don't just play slow.... there's a big difference which a lot of people dont' seem to understand.
Also slow repetitions is where you learn to relax during perfect execution of difficult material. Also vital to playing it faster.
good luck.
You need to both program good technique and push yourself physically to become a really technically proficient player. Personally I tend to push myself when I'm "playing" so I don't incorporate it into my practice.

Your mileage may vary.
I "push it" somewhat when I practice a fast lick or melody. I'll increase the tempo gradually, but I'll jump forward about 15 bpm once in a while for two reasons. The first is to see what kind of progress I'm making and how much easier playing faster has become. The second is to get mentally used to playing the lick fast.