Well to well variation in thermal cycler fluorescence?
There can be several reasons for variability and since you said that the machine has not been serviced for a long time, I guess that should be one main source of errors. Multichannel pipettes can also sometimes cause problems (based on personal experience) and it is mostly because the tips don't fit properly in all channels. 

Try varying some parameter (for e.g. concentration of RNA) and see the effect on melting temperature; if the inter-replicate variability is not high enough to make the inter-sample difference insignificant then you can continue with your experiment. Take more number of replicates to minimize variance. 

If the variability is very high then I guess you would need to get your machine repaired or use more sensitive equipments designed specifically for these kind of studies.