Vibrations in frame when pedaling Recently, when I pedal sufficiently hard, I begin to feel vibrations through the whole frame of the bike. They are not so strong as to make the bike un-ridable, but neither are they road chatter. I suspect this to be somehow related to my recent replacement of a tire (i.e. the whole tire, not just the tube), but am unsure exactly what is making the sound. 

The bike is a 1973 Sekine road bike. The frame is original. The chain and freewheel have enough stretch in them that I'm planning to replace them both again soon. The rims are true, but I don't know their age (came with the bike).

There doesn't appear to be any contact between the wheel and the frame of the bike or the brake pads. There is a periodic grating sound which also starting after I replaced the tire. Possibly it is caused by the metal plate which sits between the freewheel and the hubs, but I am not certain of this (can't replicate it when I'm not sitting on the bike). 

Anyway, this seems like a bad thing. I usually like to fix the bike myself, but I'm not sure what to fix in the this case. Any suggestions?
Well, it's hard to say from your description.  Is the vibration in time with the turning of the wheels, or much more rapid?

A vibration that's in time with the turning of the wheels may be a warped rim, a rim rubbing the brake pads, a tire that's not seated properly on the rim, etc.  (The fact that this occurred after changing the tire suggests that last possibility.)

A vibration that's much more rapid than the turning of the wheels is usually due to resonance in the frame.  You'll usually see this more often when going downhill at high speed -- the front wheel will suddenly begin vibrating violently, such that you fear losing control of the bike.  What's happening is that minor vibrations in the bike (or even just due to the road) found a resonance in the bike frame, and the vibration becomes self-reenforcing due to the way it causes the front wheel to twist.

I'd suggest first turning the bike over or hanging it somehow so you can spin the wheels freely.  Observe the wheels carefully, looking at the space between the rim and the brake pads.  Look for any change of distance as the wheel turns.  Some small amount (a millimeter or maybe two) is to be expected on a well-used bike, but any more is a problem.  Also, if the rim touches the brake pads, that's a problem.

Observe from the side and look to see whether the rim moves in and out (decreases or increases in diameter) in relation to the hub.  There should be no detectible motion in this direction. 

Next look at the tire while the wheel is spinning.  Look straight on so you can see both sides of the tire as it spins, and see if either side bulges out at any point.  Then look from the side to see if at any point the tire seems to increase or decrease in diameter.  Again, a one or two mm variation in these dimensions is probably OK, but no more.  

Also look at where the tire disappears behind the rim.  There is generally a ridge in the tire a few mm from the rim, and it should maintain a fairly constant distance from the rim edge as the tire rotates.

(Since your problem began after changing the tire, these tire checks are likely to reveal that the tire is not evenly installed on the rim.)

While you have the wheels off the ground, grasp each wheel near the rim and push back and forth sideways, feeling for any play in the bearings.  You should not feel any play.

Finally, set the bike back on the ground, grip the front brakes to lock the front wheel (leaving the rear brakes free), and push the bike forwards and backwards.  You should not be able to feel any play in the "headset" (the bearing where the fork turns).