First of all (with unplugged Uno) visually (and maybe with help of some magnifying glass) check, that really there is nothing stuck there, what should not be (tin ball, pice or wire or such) and that all copper lines on PCB are OK (and not shorted or broken). Also check all chips, resistors etc. are visually clean, good soldered and undamaged. If something looks damaged or burned, the it is probably damaged and should be replaced anyway.
If you connect it (I suggest USB charger, to be sure to not damage computer) and the main chip (MCU) is hot, I think, that it is dead.
If some other chip is hot and MCU not so much, then maybe the MCU is not dead, but something else is broken. (probably the hot part, but maybe also some resistor/capacitor shortcuted)
The way it is now makes it is dead for you anyway, so there is not much to came worse.
I would try to connect it only to USB charger (to not damage expensive computer by bad luck) and measure voltage on various points of the board (from ground). There should be 5V straight at 5V pin, also on the same 5V on ISP head, one pin of USB connector and on all power pins on MCU and the other chip between it and USB (new R3 have some atmega there too).
Take schema for that and try to measure on the right places. Be carefull to not shortcut anything with your voltmeter rod (it is easy to connect two legs with it and it may damage something not damaged yet). You can try to check voltage anywhere safely, on some places there should be 0V, on some 5V, but in default setup (no program working yet) the i/o pins are INPUT on boot, so just High resisteance and no strong voltage on them (you should get probably weak 0V until there is some interference).
If you identify (or thing so) wrong part, you can try to replace it with new one - good training even if you botch it and you risc youst price of the new part. Even the MCU can be changed with good soldering pen and some effort. The schema is simple, but the legs are tiny and many of them.
As it is effectively dead for you now, the worse what can happen is, that you burn the new part too and throw it all away (as you would anyway). The best what can happen is that you repair it and it will work flawlessly.
I build some "arduinos" from the SMD chip and other parts on home made PCBs - all worked well. So If you want some entertainment and practise, it is not bad think to try.
If you do not want risc the price of new MCU, but want the practise, yout take away and thyen resolder the (already broken) MCU - it would not repair it, but you can see for yourself, if you think you are able do similar experiments and if you enjoy it.
I found, that some chineese cheap clones have voltage stabilisers, that can break on shortcut and then do anything (usually the output is then shortcuted to input, so the power goes full where only stabilised 5V should be, but also can shortcut to ground, burn to total resistance, or something random between)
It is usually not good idea have arduino connected to two powersources (like external power source for servos AND USB for programming it) at the same time, as there may be different voltage on both and big currents between in some cases. If you need more power for servos (or something) and USB for serial/programming, then better there should be no direct connection between the external source power and Arduino +5V/+3.3V or any pin - just common ground, everything other separated by at least transistor (base on pin, collector on external source) or so.
