Must I confess with my mouth that Jesus is Lord for salvation?

No. The act of confessing Jesus as Lord was a required condition that was part of the Kingdom Gospel. Salvation in the grace gospel today simply requires you to BELIEVE that Christ died for your sins and was raised for your justification. For us today, the act of confessing that Jesus is Lord does not add one iota of value to actually save you. If it was a necessary act to save you, it would become a work, a condition, and it would diminish the 100% grace of the work of Christ on your behalf, (Rom.11:6).

The 'confession (profession)' of faith as seen in the verse below is often cited in the wrong context, and misused. The common 'sinner's prayer', and even the Roman Road salvation technique often uses this verse as instruction for a new believer to do in order to be 'saved.'

Rom.10:9-10 IF thou shalt confess with thy mouth the Lord Jesus, and shalt believe in thine heart that God hath raised him from the dead, thou shalt be saved. 10 For with the heart man believeth unto righteousness; and with the mouth confession is made unto salvation.

As mentioned above, our salvation does not have conditions to fulfil in order to merit, or earn salvation! Notice how the above verse starts. "IF you will confess...".  The word "if" is a condition. It's something required of you! The proper context of this verse in Romans is part of Paul's writings to Israel. In Romans chapters 9 to 11, Paul deals with instructions to the Jews based on their belief in the Kingdom gospel and the instructions they were required to do to merit salvation. Romans chapters 9 to 11 are not written to the Body of Christ under the grace gospel. As a matter of fact, if one had to go back to Romans 10:1, you would see exactly who the audience is that Paul was writing to during this part of his lengthy letter to Rome. 

Rom.10:1 Brethren, my heart's desire and prayer to God for Israel is, that they might be saved.

This brings in the exact context for the rest of the verses that follow. Once we recognize this, we will clearly see that these instructions are not applicable to us, but that confession of Christ as Messiah and Lord is a requirement of salvation for the Jews in the Kingdom program.

Our salvation is simply to believe! To trust that the cross-work of Christ is all sufficient to save us and justify us before God. Whether we call upon the Lord or whether we are silent in our experience, it comes down to the attitude of the heart and not to the words from the mouth.

For a more comprehensive view of this point, see here.