Xbox 360 Does not Load Games

If once you put a game in the disc tray but it's doesn't load, you still have the dashborad on the screen or its just says "Reading" then "Open tray". You'll need to check first that you didn't set your Xbox 360 to boot to the dashboard.
To check this turn on the Xbox 360 with no discs in the disc tray. Go to the System section then Console Settings then Start up make sure it's set to Disc not Dashborad.
If you select disc the Xbox 360 will play the disc when you turn it on. But if you select dashboard the Xbox 360 will stay on the dashboard when you turn it on.

If you did so and still not loading, try another thing. Make sure this issue isn't with this disc only. Try some other discs and DVDs if you have to make sure is it an issue with this disc specifically and by then you'll need to get another disc or it's the console itself.
You'll need also to make sure that your Xbox 360 is not overheating and it's in a well ventilated place.
Check that the date and time are set correctly. To check this go on the System section then Console Settings then Clock then Date and Time. If it's not set to the correct date and time, set them correctly.
The last step to try is to unplug the Hard Drive and try some discs without the Hard Drive attached just to test is it the Hard Drive or the Xbox 360 itself faulty. If you did so and it worked without the Hard Drive, it'd be obvious that it's the Hard Drive's issue. But if it didn't work it'd be the Xbox itself that needs service so you'll just call the Xbox 360 Cutomer Support to repair it for you.