No. Why? Here are two real reasons this won't take place.
1) The Channel Tunnel is a railway tunnel for high speed railroads. Railroads, and subways are different. Subway carry people to other local destinations however railroads are used to carry people from city to city, or country to country. Therefore your example is terrible.
2) A tunnel between Manhattan, and Staten Island would be so long, and so expensive that it will never work. The distance between the two islands is around 8 miles. This means that the tunnel would be as long as the Second Avenue Subway. However what makes this even more expensive is that this is underwater construction. Constructing 1 mile of underwater tunnel costs $1 billion U.S.D. which means that the tunnel you are asking to construct would literally cost $8 billion U.S.D., and to be honest no one would pay all that just for something between Staten Island, and Manhattan.
If you instead proposed a tunnel between Staten Island, and Brooklyn that did be more real, and would make more sense......