The Invisibility Cloak, as the name implies makes you invisible to sight of those around you. Can the Cloak "hide" your presence in any more significant ways?
For example, are there examples of the Cloak hiding (or, just as well, not hiding) its wearer from other senses, such as physical contact, or ambient noise, or from someone who could sense someone's body heat?
According to the "The Tale of the Three Brothers", the cloak originally belonged to Death, and he gave it up to grant the third brother's wish to "evade Death". Does that mean the Cloak is even capable of "hiding" its user from deadly effects in the environment? Could it block a spell like Avada Kedavra, or prevent physical harm (like from a fire)?