I have a love/hate relationship with the vibration device on my iPhone. I love that I can set my phone to vibrate yet I can still feel when activity happens, but at the same time, with the number of email, messages and phone calls I get on a constant basis, it tends to frazzle me sometimes. A constant buzz, buzz, buzz... whether the mute switch is on or off.
An easy way to resolve this would be to put the phone into DND (Do Not Disturb) mode, but then you wouldn't know if someone tried to call or message at all (the DND switch essentially turns off all visual indicators, including the screen turning on). There is, fortunately, another way to turn off the vibration motor, but leave the other visual indicators (i.e.: screen) on.
By default, the vibration motor is enabled at all times. To disable the motor when the mute switch is engaged, go to Settings > Sounds. You will see two "Vibrate" settings, as shown below:
My preference is to keep the vibration motor active while the phone is in ring mode, but to disable it when the mute switch is enabled. This will avoid the constant buzzing when the ringer is off, but still allow me to see when calls and texts come in. It's the best of both worlds!