Short the circuit of a voltage controlled voltage source when finding out the v(t) for \$t>0\$
Why we can see the voltage controlled voltage source as a shorted
  circuit when finding the v(t) for t>0?


First of all, it's not a voltage controlled voltage source, it's a current controlled voltage source (CCVS) since the controlling variable is the current \$i_o\$.

Now, answer the following questions:

(1) What must be the value of \$i_0\$ when \$t &gt; 0\$?

(2) Given the answer to the previous question, what is the voltage across the CCVS when \$t &gt; 0\$?

(3) Given the answer to the previous question, what can we replace the CCVS with?