Ideally, you should add DNase in not only the Trizol method, but also most other RNA extraction methods, to eliminate any DNA contamination especially if you are using your RNA for applications that require high purity of RNA (e.g. Sequencing), although some papers have stated that the addition of DNase with phenol/chloroform might reduce your RNA yield.
Some protocols might instruct you to perform RNA extraction with Trizol followed by the DNase treatment before another round of Trizol method to repurify your RNA. You can find a pretty straightforward RNA extraction protocol with Trizol in the link below.
I think that If you are going to measure or quantify gene expression of certain gene you need to use DNAse after obtaining your RNA to be sure that the positive results not due to any remenant of DNA.. you can also use RNAse treated RNA extract as a negative control to reassure that you have no any DNA contaminant..
If you are using the RNA directly in sensitive techniques e.g. qRT-PCR treatment with DNase is quite recommend. But if mRNA is purified using capture kits or similar methods, then DNase will be quite undue.