Can R visualize the t.test or other hypothesis test results?

There is a lot of things you can do. Here is just one where I draw a random sample from the standard normal distribution, then do a t-test, the plot the observed t and the t's needed to reject the null hypothesis that the mean is equal to 0.

N=20 #just chosen arbitrarily
samp=rnorm(N)
myTest=t.test(samp)
tcrit=qt(0.025, df=(N-1))

dum=seq(-3.5, 3.5, length=10^4)#For the plot

plot(dum, dt(dum, df=(N-1)), type='l', xlab='t', ylab='f(t)')
abline(v=myTest$statistic, lty=2)
abline(v=tcrit, col='red', lty=2)
abline(v=-tcrit, col='red', lty=2)

Of course your observed t will look different every time you re-run this code, which might make a good illustration if ran repeatedly.