Where on the rear of the petrol tap do 3 lines/petrol pipes from the carbs go to.

Check the Spark Plugs , Leads , Points and Distributor units are serviceable and on fully and correctly.
Go to a Motorfactors and buy some Carb cleaner.
Take off the Air pipe to the Carburettor and clean the carb as best you can with a piece of cloth , getting rid of residual oil and muck left over the years of use.. Spray the Carb cleaner in and wipe again . Start the engine without replaceing the Air pipe. Use the throttle / Accelerator with a few pumps , it will sound a bit noisey.
. Knock the engine off again and refit the Air pipe to the Carburettor. Take it for a spin on a fast road to try and clear the residual stuff from the Carb.
You can use some Petrol Additive to help clean the internal bits every few refills of petrol.

Either the float is set to high, you have trash in the lines, or the more likely scenario is the atmospheric vent line is pinched or plugged.

Reseal carb / gaskets if its from the carb body ,

Sticky float, tap the float bowl to see if it stops and frees up
Check carb bowl for dirt in float valve or wear on float valve or seat or seat o ring, make sure the float is still floating (remove and try to float in petrol) if leaking from an over flow pipe or out let

personaly id flush the carb by removing the carb including the needle[rev cable]use an old basin,take out your lowest screw releasing petrol tip the carb rotating it ending up in an upside down position petrol will come out of every in n out let,place to its original position using a bicycle pump afix firstly to your petrol inlet slowly pump air down the pump don;t pull the pump to refil remove the pump put in a blow mode an repeat 3 2 4 times,then repeat on your petrol fill pipe an repeat proseadure ,this workted 4 me goodluck

get a oil can the thumb operated type and squirt some oil down the petrol pipe and make sure pipe is upwards so the oil runs into the pump then leave for a while to wet the valves in the pump then crank engine to make sure petrol comes up or just refit the pipe and pour some petrol down the carb and watch it go

from memory
a fuel line( runs to a fitting on carbs which feeds all carbs with a common tube)
a vacumn line to opperate the fuel tap(slightly larger diameter comes off the manifold)
a tank breather line

1) This might sound daft but check if there is actually fuel in the tank. Many bikes have been stripped down by the hard of thinking only to find the tank was empty...or the tap was off... or should be on reserve... 2) Assuming this is a gravity tap (one main pipe) and not a vacuum tap (one main pipe and one smaller pipe at the side) Turn tap to off and remove pipe from tap. Turn tap on and see if fuel flows. If yes, then replace fuel pipe (this should sort it, try again with the pipe off at the carb end and see if fuel comes out. If it does but the bike still won't start then the carb jets are blocked.), if it still doesn't flow then go to3)
3) If there is fuel in the tank then remove tank and pour it out into a can. Remove tap from tank by unscrewing with the correct spanner. 4) Examine the guaze filter, it is probably full of crud and muck. Clean with fresh petrol and replace.
Hopefully that should sort it.

Check that the fuel tap is not on 'prime' (horizontal position). It should be switched to 'on' or 'reserve' (up or down for normal use. Also, check that the vacuum pipe to the carbs is not kinked. XJRs are buggers to start when the temperature is 0 deg C or less.

The thinner one is the vacuum pipe.

Small bore pipe from inlet side of no 2 carb goes to the small bore pipe on the petrol tap (this is the vacume swich pipe)
Large bore reinforced pipe goes from the large pipe on the petrol tap to the (hidden and bugger to get at) connector that points up (on the airbox side) between no's 2 & 3 carbs on the airbox
A large bore pipe goes from the rear facing connector on top of the centre cover on top of the cam cover (located under the frame beneath the petrol tank) to the connectot on the air box (this is the breather pipe)
Ensure that all pipes are hole/split free, secured with clips and that each inlet mainfold rubber (between the carb and cylinder block) has it's vacume plug fitted (located at the bottom of each rubber)