It is unlikely that hooking it up to the scope like you said would damage the pedal, and even more unlikely that shorting the output would damage it. Most overdrives only contain diodes, transistors or opamps, so not much to go wrong, although these can be killed given the right conditions. The most common faults with guitar pedals are the mechanics - connectors, switches, broken knobs, stuff that takes the abuse.
When you say "it doesn't turn on" do you mean no signal output, or no indicator LED or both? Most pedals these days are true-bypass, but are turned on by plugging in the input jack (the socket is stereo, so switches to ground). Also because of ground loops, DC jacks tend to be reverse polarity, which means if you use a DC socket that electrically connects to the metal case, the case can be at +9V rather than the 0V. I suppose that connecting to a transistors base could fry one, but again its very unlikely thats whats happened, I'm just guessing.
A schematic and maybe even a photo would be helpful
