1) Your heating system is overworked and can't keep up with the demands of your home. The old furnace is struggling and reaching the winter months, and you're noticing that it's just not keeping up with your home's needs. You can turn on the heat, and it'll put out some warmth for a while, but then an hour or two later, you've noticed that things are starting to get cold again in the rooms where your heating system doesn't run. That may be because your furnace is overworked and can't keep up with the demands of heating your home. It needs to be replaced or at least serviced.
2) Your utility bills are increasing for no discernible reason. You've had your furnace serviced recently, and it's not due for a tune-up, but yet you're seeing that the amount of heat coming out of your vents is decreasing while the amount of heat on your utility bills is increasing. That's a really bad situation to be in because it means that the furnace isn't working as well as you think. Maybe there's something wrong with the thermostat, or perhaps the filter needs to be replaced, but most likely, you need a replacement unit.
3) The temperature in your home feels inconsistent - sometimes too hot, other times not warm enough. There's just no consistency with your heating system, and it can't produce comfortable temperatures in all rooms of your home. It's not maintaining consistent temperatures, so something is wrong. It could be that you have a problem with air circulation through the vents, or perhaps one of the heat exchangers is blocked or has failed. That can lead to inconsistent temperatures. The best way to know how to fix it is to have a professional come in and assess the situation.
4) There's an unpleasant odor coming from your vents or chimney that wasn't there before (this could indicate carbon monoxide poisoning). Consider consulting with a professional if you ever notice an unpleasant odor coming from your heating system. It could be due to the fact that you have a gas leak or something is wrong with the chimney. Your chimney should never smell like smoke, but rather it should smell like soot when it's in use (that's a good thing). Any other odor that you detect could indicate a problem. And if you're smelling something foul, have it checked out right away.
5) It's getting colder outside, but you don't feel much heat coming out of the vents inside your house (this could mean that something is blocking airflow). You can tell that your heating system isn't working when it gets freezing outside, but you don't feel much heat coming out of the vents inside. The furnace might be working, but there's something blocking airflow that needs to be serviced. If you see your furnace running and yet the hot air isn't making it up into your home, then something is wrong with the ducting system, or a register may have gotten blocked.