If two viruses with totally different dsRNA invade a same host cell, will RNAi still work? 

When cell encounter dsDNA they presume it viral based and degrade it through RNAi. The Dicer enzyme cleaves short hairpin RNAs (shRNAs) into siRNA (single stranded RNA). The RISC complex binds the siRNA and guide it to complementary mRNA strand where it gets cleaved and eventually degraded. This is how cells degrades viral dsRNA.

RNAi will still work for multiple viruses invading a single host cell since multiple dicer proteins recognizes dsDNA.

^ Lee, Young Sik; Nakahara, Kenji; Pham, John W; Kim, Kevin; He, Zhengying; Sontheimer, Erik J; Carthew, Richard W (April 2004). "Distinct Roles for Drosophila Dicer-1 and Dicer-2 in the siRNA/miRNA Silencing Pathways". Cell . 117 (1): 69–81

9 hours ago, Vexen said:

When cell encounter dsDNA they presume it viral based and degrade it through RNAi. The Dicer enzyme cleaves short hairpin RNAs (shRNAs) into siRNA (single stranded RNA). The RISC complex binds the siRNA and guide it to complementary mRNA strand where it gets cleaved and eventually degraded. This is how cells degrades viral dsRNA.

RNAi will still work for multiple viruses invading a single host cell since multiple dicer proteins recognizes dsDNA.

^ Lee, Young Sik; Nakahara, Kenji; Pham, John W; Kim, Kevin; He, Zhengying; Sontheimer, Erik J; Carthew, Richard W (April 2004). "Distinct Roles for Drosophila Dicer-1 and Dicer-2 in the siRNA/miRNA Silencing Pathways". Cell . 117 (1): 69–81

Thanks a lot for your explanation! Very helpful!