No, speaking in tongues is not necessary for salvation nor is it vital to spiritual fruitfulness and deep fellowship with God. The apostle Paul tells us that speaking in tongues is the least of all gifts, which quite flies in the face of those who push it as a necessary spiritual skill.
1 Corinthians 12:28-30
28 And God has appointed these in the church: first apostles, second prophets, third teachers, after that miracles, then gifts of healings, helps, administrations, varieties of tongues.
Paul places "varieties of tongues" at the very end of his description of the hierarchy of spiritual gifts. Clearly, it is not the vital spiritual gift that so many in the Pentecostal and Charismatic communities tout it to be. In fact, Paul goes on to ask a series of rhetorical questions that highlight the fact that believers do not all obtain the same spiritual gifts:
29 Are all apostles? Are all prophets? Are all teachers? Are all workers of miracles?
30 Do all have gifts of healings? Do all speak with tongues? Do all interpret?
Paul has answered his questions here earlier in chapter 12 when he wrote,
1 Corinthians 12:18-19
18 But now God has set the members, each one of them, in the body just as He pleased.
19 And if they were all one member, where would the body be?
Paul asks how a body could be a body if the body consisted of just one part of the body. By definition, a body is made up of many different parts. So, too, the Body of Believers is made up of members who possess different spiritual gifts. Not all are teachers, or healers, or prophets, or speakers in tongues. If all believers had the very same gifting, where would the Body be? How could it function? Plainly, the answer is that it couldn't. So, those who press you to speak in tongues and tell you it is necessary to full spiritual living are directly contradicting Scripture.
Paul is quite aggressive about downplaying tongues.
1 Corinthians 14:5
5 I wish you all spoke with tongues, but even more that you prophesied; for he who prophesies is greater than he who speaks with tongues, unless indeed he interprets, that the church may receive edification.
1 Corinthians 14:9
9 So likewise you, unless you utter by the tongue words easy to understand, how will it be known what is spoken? For you will be speaking into the air.
1 Corinthians 14:12
12 Even so you, since you are zealous for spiritual gifts, let it be for the edification of the church that you seek to excel.
1 Corinthians 14:19
19 yet in the church I would rather speak five words with my understanding, that I may teach others also, than ten thousand words in a tongue.
Selah.