Check the high limit switch. If the furnace is allowed to continuously over heat, It's likely the high limit switch has burned out.
Make sure that the outlet vents are clean, and clear of debris. You need good air flow both in, and out of the system. also make sure the heat exchanger is clean, so it can transfer heat properly to the passing air.
If the thermostat has an option for Fan, switch it to Fan and make sure the fan starts. If the fan does not turn on, you should contact a trained HVAC tech. Capacitors can be deadly if not handled properly. You could accidentally discharge the capacitor into yourself, leading to electrocution. Or short the capacitor, leading to an exploding capacitor. In either case, it could be deadly.
Your furnace is shutting down on limit. There can be a number of causes for this.
Start by making sure you have a clean filter in the furnace. A dirty or restrictive filter can cause this issue. Alternatively you could also have a dirty or plugged up A/C evaporator coil which is located above your furnace. This can also cause a furnace to cycle off on high limit, however is uncommon unless you have run your furnace without a filter for prolonged periods of time. Also if you have a high efficiency furnace, there is more of a chance to have the secondary heat exchanger plug up rather than the A/C coil.
You can also have a blower motor issue. The motor can be faulty or seizing up causing it to not spin fast enough to move enough air through the furnace. Another cause may be that you have a faulty limit that is opening prematurely.
The major issue that can cause this problem is a heat exchanger issue. If your furnace is cycling off on high limit, and everything checks out, you may have an internal blockage of the heat exchanger. If this is the case either the heat exchanger or the furnace will need to be replaced. A combustion analysis of the furnace is required to prove this is the issue.
If you happen to have a 2 stage high efficiency furnace you could have a bad ventor motor gasket which is allowing air to be drawn in behind the motor. Big cracks in the condensate collector pan can also cause this but is very uncommon. You can also have the a two stage gas valve high fire solenoid stuck open, but this is also very uncommon and can only happen on 2 stage furnaces. Also if you have a bryant, payne, or carrier high efficiency, you could have a cold spot baffle leakage that is allowing air to be drawn in between the primary and secondary heat exchanger however this is also very uncommon.
If you think your blower is not spinning fast enough you may also have a bad blower motor, or if an ecm motor, the control board may be bad. If its a fixed speed motor and your getting power to it but its not spinning fast enough it could possibly be a bad blower motor. If you decide to change out the blower motor, make sure to get the appropriate replacement capacitor to go with it. I actually did have the same issue on an older carrier high efficiency furnace a few weeks back. It ended up being a bad blower motor, however the motor spun freely and had no indications of seizing or hard starting.
Best Answer
Your high limit switch may be bad, and need to be replaced.
Your furnace may be overheating, and the limit is not reacting. So the fusible link is doing just what it's supposed to.
There are a few things that can cause the furnace to overheat, a dirty filter being the number one cause.