Does John 3:5 teach that baptism is necessary for salvation?

I think that “born of water” refers to the “washing of water by the word” (Eph 5:26). Note that Peter refers to being “born again... through the word of God” (1 Peter 1:23), the very thing John is speaking about in these verses (cf. John 3:3, 7). 

No, baptism isn't needed to be saved. John 3:5 isn’t talking about baptismal regeneration. Salvation is by grace through faith and not by works of righteousness (Eph 2:8-9; Titus 3:5-6). NG & TH

Water baptism doesn’t seem to be what Peter has in view in 1 Peter 3:21. The English word "baptism" is simply a transliteration of the Greek word baptizo (βαπτίζω), which means "to immerse." Baptizo does not always refer to water baptism in the New Testament (cf. Matthew 3:11; Mark 1:8; 7:4; 10:38-39; Luke 3:16; 11:38; 12:50; John 1:33; Acts 1:5; 11:16; 1 Corinthians 10:2; 12:13).

So Peter is not talking about immersion in water, as the phrase "not the removal of dirt from the flesh" indicates. He is referring to immersion in Christ's death and resurrection through "an appeal to God for a good conscience," or repentance. Again, it is not the outward act that saves, but the internal reality of the Spirit's regenerating work (cf. Titus 3:4-8). JM
Note that Titus 3:5 or its context is used at least 6 times among these answers here.