What is an evil heart of unbelief?

The phrase you quoted comes from the letter to Hebrews:

Heb. 3:12 Take care, brethren, that there not be in any one of you an evil, unbelieving heart that falls away from the living God.
Heb. 3:13 But encourage one another day after day, as long as it is still called “Today,” so that none of you will be hardened by the deceitfulness of sin.

In this context specifically, the writer is describing the heart of a believer who has moved away from a faithful walk with the Lord and has begun to live in the flesh. They have allowed sin to harden their heart to the leading of the Holy Spirit, and therefore they are in danger of experiencing shipwrecked faith. 
The writer is not describing a loss of salvation, but a loss of fellowship and potential reward. For a complete explanation of this passage, we recommend you listen to our Hebrews Bible study.