Can you get in big trouble if you sit in a first class plane seat when you are economy class? ?

One, they have a list of names and seat assignments 

You would be escorted back to your assigned seat

You would be arrested upon arrival

You could be charged with theft of service

Yes you would be banned from that airline

How can you lose your ticket between the gate and the plane? When you get to the plane you have to show your boarding pass which has your seat number on it. Plus the First Class attendant knows how many passengers they have in their section so you would be spotted in no time and asked to move to your proper seat.

the cabin attendant has a list of all open seats and will tell you to move back where you belong.  S/he can also look up where your seat is, but may insist on seeing photo ID at that time.  You can refuse to comply -- and be removed by police.  Taking that seat is theft of service unpaid for.

You would be asked to move and shown your seat you paid for........ if you refused to move you would be removed from the aircraft

Usually the flight attendant will escort you to the appropriate section. They have a passenger manifest, so they can find out where you should be

If you lost your ticket before the gate, you wouldn't be able to get on. If you lost your ticket after the gate, then they would have your name, boarding pass, and seat assignment in their computer. As soon as they realized you didn't belong in First Class (I'm guessing not long) they would look up your seat, and politely ask you to go to it. If you refused, they'd stop the plane and let you off into custody.

They could kick you off the plane.  Usually this is done before the plane leaves the airport, but it could be done in international airspace.