Is it safe to use a Tamron lens with my Canon EOS 600D DSLR? Will it affect the sensor in any way?
Third party lenses are generally safe to use. The only situation that is likely to damage the camera is if the lens focuses light weirdly and so creates a part of the camera a lot hotter than usually or overloads the sensor. This is very unlikely. Third party lenses (especially from manufacturers such as Tamron, Tonkina and Sigma) are usually of equal (and in some cases better) quality to their counterparts in the camera manufacturers line up.
As long as the lens has the appropriate mount for your camera then it will not damage your camera, and if it does it is a major defect. These defects occur at a rate that is many, many, many times lower than the rate of damage from other sources. Probably a similar rate to that of the original manufacturers lenses.
No, using a third party lens will not damage your sensor in any way. There is a large air gap between the lens and the sensor, so there's no way that using any lens can damage the sensor.
(This assumes that you're using a lens for the correct lens mount. If you're using something for a wildly inappropriate lens mount and forcing the lens onto the camera somehow you could conceivably do some damage, but if you're forcing things onto the camera, you deserve everything you get).