Just some ideas off the top of my head:

1. Mutation-to-expression modelling: For each mutation, test it's association to the expression of differentially expressed genes (DEGs) in the mutation's 'vicinity'. This can be as easy as building a linear regression model with expression as the y (dependent) variable and mutation present/absent as x (predictor). From this, you could derive R-squared values and cross validated 'shrunk' R-squared values, along with p-values. y variable would be continuous; x variable would be categorical with mutation absent as reference/base level.
2. Transcription factor binding sites: Check for new TFBS (transcription factor binding sites) that may be introduced as a result of each mutation. Look at databases like JASPAR to do this - there are also other threads on biostars. There are undoubtedly some mutations in your data that are going to modulate expression of nearby genes. For an idea of mechanism, see the wonderful study by Manour: An Oncogenic Super-Enhancer Formed Through Somatic Mutation of a Noncoding Intergenic Element
3. Histone binding regions: Check for overlapping histone methylation (e.g. H3K27me3) and acetylation (e.g. H3K27ac) binding regions - this data is available from the UCSC, as far as I know. A mutation in such regions could modify chromatin structure and alter expression.
4. Transcription start sites: Overlapping transcription start sites (TSS) - again, available from UCSC I believe
5. In silico prediction: Use one of those functional / pathogenicity prediction tools. There have been many tools released in recent years, including ones tailored for cancer and somatic mutations. Take a quick look here: A: pathogenicity predictors of cancer mutations

Noe that, technically, you could introduce all of the data from points 2-5 into the model mentioned in point 1. This would then be a robust way to assess the role of each mutation in relation to gene expression.

Finally, thinking just about the RNA-seq data, you could deconvolute it in order to identify immune cell-types that may be present in the tumour. This would give you an indication of the amount of immune cell infiltration, which is likely to differ across your tumors.

There are yet more ideas that I have not mentioned.

Kevin