Is it possible to install an app and exclude some of its requested permissions? Occasionally I want to install an application which requires certain permissions that I feel it really shouldn't need. Can I install the app without giving it these permissions? For example, GPS location, phone state, internet connection, etc.

Is it possible to change the permissions of an application once it is installed?
Prior to Android 4.3 (Jelly Bean)

No. Without a custom ROM it's an all-or-nothing affair. Which is part of the reason Google encourages developers to ensure that they're asking for the absolute minimum permissions required for the app to work.

Short of getting the app code and modifying it, you either need to accept the access request or not use the app. Some alternatives exist if your device is rooted.

Android 4.3/4.4 (partly also 5.x)

There's a semi-hidden permissions manager (App Ops) that can be reached through a third-party app, but doesn't require root or a custom ROM. Android Police has covered it here. 

Note, however, that Google has said that this was released in error, and newer versions of Android have once again disabled this.

Android 4.4+

Additionally to above mentioned App Ops, here you can also use the Xposed Framework (for Android 5+ see here) with modules like Xprivacy or Protect My Privacy. While this will require your device to be rooted, it offers additional functionality.

Android 6+

marcioggs outlined functionalities provided in Android M in a separate answer.