Hi @Robyn Henderson,
What your tenant does is a regular part of maintaining the boiler system in old buildings that have steam heat. Although it's recommended to do it weekly, skipping a week or 2 shouldn't cause a major issue UNLESS you have another issue going on. I agree that 60 years is a very long time for a boiler, however as the saying goes, "they don't build 'em like they used to." Here are a few thoughts in no particular order.
1. Do not do ANYTHING to the boiler if it's working now. Start pricing your options around March/April after heating season is over. If it ain't broke, don't fix it.
2. As a 3 family you have a lot of flexibility in what you can do with the heating system. Gas vs oil. Steam vs hot water. If you have equity in the property and want to upgrade the property to increase rents you could install separate heating unit for each apartment. That will reduce your operating expenses too. It's extremely popular now to forgo boilers altogether and install dual-purpose HVAC units that both provide cool and hot air. Just make sure the units are properly insulated.
3. Check your PM for an addition concern.
-Ahmed