Can't run TimeMachine.app

Exactly what does happen when you try to open Time Machine? Are you clicking the Dock icon or starting it from the Applications folder?

Then there is something amiss if it doesn't even bounce or provide an error alert. So, I think you are left with doing the following:

Reinstalling Lion/Mountain Lion Without Erasing the Drive

Boot to the Recovery HD: Restart the computer and after the chime press and hold down the COMMAND and R keys until the menu screen appears. Alternatively, restart the computer and after the chime press and hold down the OPTION key until the boot manager screen appears. Select the Recovery HD and click on the downward pointing arrow button.

Repair the Hard Drive and Permissions: Upon startup select Disk Utility from the main menu. Repair the Hard Drive and Permissions as follows.

When the recovery menu appears select Disk Utility. After DU loads select your hard drive entry (mfgr.'s ID and drive size) from the the left side list. In the DU status area you will see an entry for the S.M.A.R.T. status of the hard drive. If it does not say "Verified" then the hard drive is failing or failed. (SMART status is not reported on external Firewire or USB drives.) If the drive is "Verified" then select your OS X volume from the list on the left (sub-entry below the drive entry,) click on the First Aid tab, then click on the Repair Disk button. If DU reports any errors that have been fixed, then re-run Repair Disk until no errors are reported. If no errors are reported click on the Repair Permissions button. Wait until the operation completes, then quit DU and return to the main menu.

Reinstall Lion/Mountain Lion: Select Reinstall Lion/Mountain Lion and click on the Continue button.

Note: You will need an active Internet connection. I suggest using Ethernet if possible because it is three times faster than wireless.

Please try each of the steps below that you haven't already taken.

Step 1

From the menu bar, select  ▹ System Preferences ▹ Time Machine and make sure TM is ON.

Step 2

If you're backing up to an external hard drive attached to your Mac, open the backup volume in the Finder. You should see a folder with the name "Backups.backupdb". You should not see a file with a name ending in "sparsebundle". If you do see that, the backup was created over a network and can only be used that way.

Step 3

In the Finder, select Go ▹ Go to Folder from the menu bar, copy the text on the line below into the box that opens, and press return:

~/Library/Preferences/com.apple.finder.plist

Move the selected file to the Desktop, leaving the Finder window open for now.

Relaunch the Finder by selecting  ▹ Force Quit from the menu bar, then selecting Finder and pressing return. Your Finder settings will be lost.

Try again to enter Time Machine. If you still can't, put back the file you moved to the Desktop, replacing the newer one that will have been created in its place.

Otherwise, close the Finder window you opened and delete the file you moved to the Desktop. Recreate your Finder preferences.

Step 4

Launch Disk Utility and select the icon of the backup drive in the pane on the left (not any of the volume icons below it.) At the bottom of the window, you should see this:

Partition Map Scheme : GUID Partition Table

If the partition scheme is anything else, the drive has to be repartitioned (with the loss of all data.)

Step 5

Boot in safe mode by holding down the shift key at the startup chime. Try again to enter Time Machine.

Note: If FileVault is enabled under OS X 10.7 or later, or if a firmware password is set, or if the boot volume is a software RAID, you can’t boot in safe mode.

Step 6

Launch the Console application in any of the following ways:

☞ Enter the first few letters of its name into a Spotlight search. Select it in the results (it should be at the top.)

☞ In the Finder, select Go ▹ Utilities from the menu bar, or press the key combination shift-command-U. The application is in the folder that opens.

☞ Open LaunchPad. Click Utilities, then Console in the icon grid.

Click the Clear Display icon in the toolbar. Try again to enter Time Machine. Post any messages that appear in the Console window – the text, please, not a screenshot.