Matthew 25:41 speaks of "everlasting fire" for the wicked. Does it go out?
Yes, according to the Bible, it does. We must let the Bible explain itself. Sodom and Gomorrah were destroyed with everlasting, or eternal, fire (Jude 7), and that fire turned them "into ashes" as a warning to "those that after should live ungodly," 2 Peter 2:6. These cities are not burning today. The fire went out after everything was burned out. Likewise, everlasting fire will go out after it has turned the wicked to ashes (Malachi 4:3). The effects of the fire are everlasting, but not the burning itself.
Doesn't Matthew 25:46 say the wicked will receive "everlasting punishment"?
Notice the word is punishment, not punishing. Punishing would be continuous, while punishment is one act. The punishment of the wicked is death, and this death is everlasting.