Today my SP3 wouldn't turn on. I did the usual hold power button for 30 seconds and after a few attempts of this it powered on. Happened again later in the day only this time it took longer. Now it won't respond to any of the advice on Microsoft's help page. The only thing else I can find online is try freezing it for a few minutes. 
What should I try next? The damn thing is a few months out of warranty so thanks Microsoft for only giving a measly 12 months on that.
First off, don't be an idiot freezing it. That won't do anything.
Take it apart and inspect it for water damage on the logicboard. That's one of the most common things to kill any sort of electronic device (aside from PSU's).
Does the device charge? How do you know the charger works?
Do the SP3's turn on without a battery? Try taking out the battery.
Is there obvious damage to the charging port or charger?
Not sure what you can do as you cant really fix this type of device. Did you install the firmware update for the battery problem?
I don't believe so. I managed to get it up and running again holding the power button in. I'm not sure why it seems to be happy to turn on after leaving it for some time. A battery issue would not surprise me in the least, everything else seems to work fine.
Don't worry, I would never try that. 
I don't know if I trust myself taking it apart. I don't think there's any water damage, I never have liquids near it.
I have managed to turn it on again without charger plugged in. I expect the issue to happen again in the near future if the rest of today is anything to go by.
I have noticed no damage to any parts, however when charging the indicator light does switch off and then on every 10 to 30 seconds or so.
Moisture can kill devices, and just because you don't think water got down there, doesn't mean it hasn't.
Since it appears to be working for the most part, my guess would be battery (because of the "flickering" charging light), or the actual power button being damaged. Both repairs involve you actually taking apart the device.
Edit: or still something wrong on the logicboard, like an intermittent issue...
sounds like the battery or the connector between the battery and the main board, or something going bad in the power section of the main board (in that order of probability). you can check the battery connector as it may be loose or damaged/frayed and you can open it up and inspect the main board and look for damage and swollen capacitors. if i were you i would check the connector then replace the battery to see if that solves your problem before opening it up. even ifixit couldnt take it apart without breaking it so this is far from easy or simple and you will likely damage it by opening it (by design fuck you microsoft)
https://www.ifixit.com/Teardown/Microsoft+Surface+Pro+3+Teardown/26595
I think I'm just going to send it to Microsoft. I don't want a cracked screen!
Sorry the update is coming so late. I just wanted to let those involved in this thread know I ended up doing nothing and the Surface is now and has been running perfectly fine without me doing anything.