So if kind of depends on what you are working with. If you are collecting viral supernatant in vitro, then no you wouldn't have to because all that will be in the supernatant is the viral progeny and things that the host gives off like cytokines, etc depending on what you are working with. If you are collecting cell pellets as well then that is when you will run into problems with host genome but again you should be looking for viral progeny which will be released into the supe. If your virus is killing off the host cells some genome will be in the supe as well, but with all of that said, you should be using primers that are specific to your virus so if you check that sequence in a program and double check that the primers don't also pick up anything in the host genome, which they shouldn't, then regardless of anything you should only be picking up your viral genome. 

Also, you could always do a DNAse step in the RNA isolation and not worry about host DNA genome contamination.

When I used to work on RNA viruses we would take the supernatant, extract the RNA and run qRT-PCR and that was all. 

Hope that was helpful, good luck.

Possible: Maybe (in some cases)

Advisable: Definitely not

The extraction/purification process has several processes that increase the potential to detect viral RNA including 1) Lysing cells to release viral RNA 2) Removal of cellular components that can inhibit PCR polymerases such as haemoglobin and 3) Maximise recovery through precipitation. As and added benefit, the chemicals used to lyse cells typically inactivate the virus you are trying to detect (and others that may be present in the sample such as HIV, Hep B etc) thereby reducing the risk of exposure to the operator

It may be possible to amplify viral RNA directly form blood samples without extraction/purifiation but you'll experience significant false negativity and possible exposure of the operator to any pathogen present in the blood. Also, if you are using an internal control for QC purposes you'll have to repeat many/most samples due to polymerase inhibition which will quickly be more expensive and time consuming than performing the extraction on all samples

There are ways of increasing the recovery of RNA without using silica colums, but this is the easiest format available at present. and is extremely efficient at purifying the target viral RNA

All the best

Thank you all for your replies.