Off the top of my head, here goes...
Sequence of normal operation will be something like this.
A)thermostat closes, sending 120v to draft fan; fan starts
B)draft proving switch closes, sending 24v to ignition control
C)ignitor begins to spark
D)gas valve opens
E)~5 seconds for flame stabilization
F)flame rectification looks for microvolt signal from flame rod
G)unit runs until thermostat opens
Some units with pilots are slightly different:
A)thermostat closes, sending 120v to draft fan; fan starts
B)draft proving switch closes, sending 24v to ignition control
C)ignitor begins to spark
D)pilot valve opens
E)~5 seconds for flame stabilization
F)flame rectification looks for microvolt signal from flame rod
G)ignition control recieves flame rod signal, opens main burner valve
H)unit runs until thermostat opens
Cheapest thing to do first is clean the flame rod and ensure the pilot assy ground is intact and not corroded. The ground is very important as the flame rectification signal/voltage is actually the electrical difference between the pilot housing and the flame. Flame rod systems are nothing like a thermocouple or powerpile, as the later generate voltage, the flame rod is a voltage sensor. Any slight ash or corrosion will insulate it. To clean a flame rod, sanding with a crisp dollar bill is recommended, although I have used a wire brush everso carefully. Do not use sandpaper or sandcloth as any residual adhesive or sand can insulate the rod when heated.
Symptoms of a dirty flame rod would be...ignition of pilot and failure within ~5 seconds.
A restricted pilot burner orfice and poor gas supply can also be a possibility. Looking at the pilot and how it changes when the burner is on is what to look for here. If the pilot dwindles down to next to nothing when the burner is running clean/replace it.
If the burner runs a bit and then fails, chances are the inducer motor may be suspect. Motor can loose a bit of rpm performance as they age. Hot air is denser than cold, and when combined with a weak motor, burners can fail on the draft proving switch opening. A usual symptom of this is lack of capacity of hot water/lukewarm and frequent, short cycles of heater.
Thermostat... well the draft fan would never start.
Coil/s in gas valve are a possiblity for failing, although more remote of a problem.
Weak spark, replace ignition control/spark wire. You should have a pretty good gap that the spark covers. Also, if spark color is yellow/orange its weak. You want to see a white and a bit of blue. Sparking sound that is irregular in rhythm usually indicates a faulty wire.
Check all spade connectors. Make sure they are tight and mating surfaces are not corroded.
Also check venting for any backpitch/sags or blockage.
Good luck. Now you see why most guys just sell 'em a new one.