I have an old house and don't have a window to open for the portable AC to vent to. Can I vent it out through a heat vent (my home has modern ventilation)?
You will be pushing humid warm air into the ductwork. That is not a good plan for a number of reasons.
1) This is essentially the same as venting the portable ac into the house since the ductwork feeds air to the house. The ac unit is trying to remove heat and humidity from the house to cool it. That is why the vent needs to exit the living space.
2) introducing moisture into the ductwork will cause mold to grow in them and may cause them to rust if they are metallic.
3) The humid air pushed into the duct work will find its way to the expensive parts in your furnace and damage them as well.
I would try thinking of it like a clothes dryer that you need to vent to the outside of your space. Could you add a duct through a wall with a weather cap on the outside termination?
Kris's answer says that the exhaust air of a portable AC unit is humid, but that is incorrect. Kris's answer is correct for AC window units, which definitely direct both the heat and condensate drain directly outside, but the question was specifically asked about portable AC units.
The condenser coil's (on the hot side, typically vented to outdoors) temperature is always going to be equal to or greater than ambient temperature. This means that the temperature of the condenser coil is never below the dew point of the air, so there is no condensation to deal with.
The evaporator coil's (on the cold side, typically vented indoors) temperature is always going to be equal to or less than ambient temperature. When the evaporator coil temp is lower than the dew point of the air going through it, water will condense and the unit will either collect it to drain later, or you can plumb it directly to the drain.
A portable AC unit is essentially just a portable dehumidifier, but with improved and separated venting for both the hot and cold sides. The exhaust of a dehumidifier is not moist, because the condensate is collected/drained. On the same note, a portable dehumidifier also creates heat from the phase change process, and if you vent a portable AC unit indoors you are essentially doing the same thing. You'll lower the humidity of the entire building, but one room will be cold with the rest being slightly warmer.
So to answer your question, yes you can vent a portable AC unit wherever you want to indoors, as long as you don't care about pushing heat there. Ideally you would vent it outdoors, but second best would be venting into a return air vent and running the HVAC furnace circulation fan but without turning on the heat in your furnace. This will spread the heat around enough to not be as noticable.
Dumping heat into a heating vent will push hot air into wherever the next closest heating vent is on the same supply line. If this is a bedroom right next to this room, it's going to get pretty hot in there.
I've helped with cooling a server room in commercial office space. Portable AC units are used in situations where mini split AC units aren't in their budget or they're not allowed to make penetrations in the exterior building envelope. Typically they vent the exhaust into the plenum above the dropped ceiling tile, which is the return air in most offices. Yes, it heats up the rest of the building, but it makes the one room colder than the rest as intended.
Ideally they would have just vented the room better and skipped the portable AC unit, but the IT guy in charge thought that servers would be too hot at 75F ambient and that the room needed to be 60F instead. I disagreed, but it's not my server room or equipment either.
Either way the exhaust is hot and dry, not hot and humid. Dump it anywhere you want, it's definitely not like a clothes dryer at all. You won't have a mold problem. Go for it.