The axis limits are being set using the plot.ts(). So if you want to override the data, you can either set in there or before the par() function.
Though plotting behavior can be unexpected. below are the two options you can try:
Option ( 1 ) :
TimeSeriesData <- ts(new$Value, start = c(input$years[1]), end = input$years[2])
plot.ts(TimeSeriesData, ylim = c(<your min>, <your max>))
abline(h=amount$toxicityLevels, col="blue")
Option ( 2 ) :
TimeSeriesData <- ts(new$Value, start = c(input$years[1]), end = input$years[2])
par(ylim = c(<your min>, <your max>)
plot.ts(TimeSeriesData)
abline(h=amount$toxicityLevels, col="blue")