In a word: never.
But slightly more usefully, here's my 50øre. If you publish a paper that depends on the result, are you going to be embarrassed if the referee says, "Can you clarify your use of Theorem X?". If you feel happy saying, "A,B, and C all published result depending on it, so I figured I was safe." then go ahead. If you're not quite so sure that A, B, or C check things quite so carefully as you do, check it yourself.
So, for example, if it's a result about differential topology on loop spaces then I would check it very carefully because I ought to know about that stuff and I would be embarrassed if the referee said that. But, say, Kuiper's result on the contractibility of the general linear group, then I figure it's not quite my area of expertise and plenty of other people have used that result in the meantime that if someone finds a mistake now then my minor embarrassment is going to vanish into nothingness besides the other things that are going to come crashing down.
To put it a slightly different way, suppose that you prove X, which depends on Y. Then someone proves W depending on your X. Later, Y is found to be false. When you and the person who proved W happen to be at the same conference, do you a) hide in a corner and hope that they don't see you, or b) go to the pub and have a good laugh about it all. If you think it'll be (a), then you should have checked Y. If (b), then you're in the clear.
Best Answer
I don't think academic math comes up with general policies for things like this. The dreaded "common sense" should be applied. How well known is the old result, for example? Some things are very old but everyone knows them.
Or..to give an extreme example: Suppose your result was little more than a corollary of an old obscure result. Writing up the old result in new language and then tacking your result on the end wouldn't look too good... So in such a case I would not include the proof of the old result and just accept that I had a very short paper.
The other extreme is that the old obscure result is fairly short and not too hard anyway, in which case people would believe you/be able to read the old paper if they really had to... so again I would not include the proof of the old result.
If you judge to be somewhere in between: that the old result is very relevant and is hard or subtle in some way but that doesn't overshadow your own work, then people will probably just be pleased to see a nice account of it.