Hey, Steve and Clayton!!
Glad to have you here!
My first guess is that the carb bowls are gunked up with old gas. There are 6 slotted screws on the bottom of each carb that hold the bottom cover (chamber) on. Be careful of the gasket.
If you take it apart, you will probably start with the top cover (Dashpot), 4 slotted screws. Again be VERY careful when you remove this cover as you can tear the rubber diaphragm inside. Please note that the diaphragm has a dimple along the edge that fits in a space on the edge of the metal casting. There is a long needle attached to the bottom of the air valve inside the casting. Be careful not to bend the needle as this meters the gas into the airflow.
Did you see the funny plastic knob (dashpot damper) at the top of the housing? You have probably already pulled it out and wondered why that brass sleeve was there. Well, it acts like a shock absorber for the air piston. As the engine is running each piston pulls in air on the intake stroke. You do not have a smooth flow of air into the engine, but bursts of air coming in. This causing a fluctuation of the vacuum in the carb. This fluctuation will cause the air valve to vibrate up and down, and then would constantly be changing the gas flow into the mixture.
We stop this vibration by keeping some lite oil in the tube (air valve guide rod, AGVR) at the of the air valve and that brass tubing with it's retaining washer slows the movement of the oil in the tube. Now you don't want to keep pouring oil it there until it is full. You just want to fill the tube(AVGR) about 1/2 way full as the brass tube will displace some of the oil. I always used a chemistry wash bottle to fill the tube.
Anyway, I digress! Get it all clean. As other here have already suggested, get a small gas can and put a rubber hose from the can to the fuel inlet of the carbs. Put the can high so that gravity can do it's thing and try to start the car.
Check out some of my sites below. Download the file "TR7man.pdf" from my download page.
We will be here is you need us!
jclay
[url="http://web.mac.com/jclaythompson/iWeb/Site/Welcome.html"]My Triumph Site[/url], [url="http://web.mac.com/jclaythompson/iWeb/Technical/Intro.html"]Technical Stuff[/url], [url="http://homepage.mac.com/WebObjects/FileSharing.woa/53/wo/HJMTK8gsojtwKleP.1/0.2.1.2.26.31.97.0.35.0.1.1.1?user=jclaythompson&fpath=Triumph_Articles&templatefn=FileSharing4.html"]Download Page[/url]