Old Answer:
I'm having the same problem (with ADB in Linux Mint 17, 32bit Quiana). I made sure that the file ADB in platform-tools directory has executable permission but it appears that will not run at all (even if I try stopping Android Studio and running ADB on its own). I tired uninstalling and re-installing the Android SDK Platform Tools (v 23.1) but that didn't help. Is there a way to revert back to an earlier version of Platform Tools?
Improved Answer:
If you're using a 32-bit Linux system to run Android Studio and have had the misfortune to have updated to platform-tools 23.1.0 you need to revert back to platform-tools 23.0.1.
Here's how I did that:
Download the 23.0.1 files with this link: https://dl-ssl.google.com/android/repository/platform-tools_r23.0.1-linux.zip
Move the downloaded zip file to the directory where your Android SDK is installed (something like ~/Android/Sdk).
Rename the existing directory containing the 64-bit files ( ~/Android/Sdk/platform-tools to ~/Android/Sdk/platform-tools-64-bit ). Note: you could just delete these files but I saved them. They don't take up much space.
From the ~/Android/Sdk directory unzip the downloaded zip file. That should have created a new directory ~/Android/Sdk/platform-tools containing the platform-tools 23.0.1 files and sub-directories that you need for your 32-bit system. ADB should now work just like it did before this "update" broke it. (Bad Google, Bad, Bad)
Reference:
https://code.google.com/p/android/issues/detail?id=196866