Your furnace definitely has a filter. To access the filter compartment remove the lower of the two metal panels on the front. The filter location is horizontal at the very bottom in the plane of the very bottom of the furnace. It will probably have a metal loop resting on it to keep it in place in the upward air stream. You may want to remove that metal loop to more easily put in the new filter, but it may be less trouble in the end to work around it.
(The upper panel is for inspecting and servicing the burner. This has nothing to do with the filter and you do not want to remove this.)
Before removing the panel turn the furnace off at the thermostat and at switch it off inside the closet if there is a switch there. Do not attempt to remove the access panel with the fan running. Carefully inspect the way the lower panel fits so you can return it to that state. You may want to have a new filter in hand so you can put it in right away.
To remove the panel grasp the top edge of the panel labeled TAPPAN Air Conditioning and move the panel enough to free it by pulling up, then work the panel out. It is a little tricky until you get experienced at it. The metal edge might be sharp. You might want to wear gloves until you are practiced at it. When you remove the panel you will see the squirrel-cage fan and the filter will be below that.
Using the standard grade of filter you might want to change the filter once a month or two or three. It just depends on how the filter is performing in your house. Here is a video showing the same type of furnace as you have, but I think you do not have to remove the upper access panel like this guy did. https://www.youtube.com/watch?v=cZknWY1zAUk
Our house is 45 years old and we have lived in it 39 years. Ours is a Carrier natural gas furnace of the same configuration as yours, but at installation 25 years ago I opted for an extra filter unit under the furnace which houses a 6-inch thick pleated paper filter. These are $25 to $30 each and I change it once a year. It is accessed through its own panel so it has been a long time since I opened the regular access panel.
The original furnace was of the same type and when we moved in the filter had fallen down and was lying on the bottom of the plenum below.
Best Answer
Do you mean HRV/ERV? They won't help with allergens. In fact they are likely to make the situation worse. Their job is to bring in fresh air and pre condition it with the outgoing air. They do have filters but they are just to protect the exchange core. If your allergic reactions are due to pollen and the like an HRV will make it worse as the equipped filter is not a MERV rated filter. You would be better off installing a whole house HEPA unit.
Duct mount HEPA
Elecro-Air Five Seasons
I've installed lots of these. You would leave the furnace fan running constantly as well as the HEPA filter fan. There are much better options available from IQAir but they are very expensive.