"The sum of two rational numbers is always rational." - true, for example, 2+0.5=2.5 (decimals and fractions are rational)

So statement is true.

"The product of a nonzero rational number and an irrational number is always irrational." - true, 5*√2=5(√2) remains irrational. But watch 0*√2=0 (rational). However, 0 is not "nonzero", so the statement remains true.

"The product of two irrational numbers is either rational or irrational." √2*√3=√6 - irrational

√2*√2=√4=2 - rational.

So statement is true.

Finally,

"The sum of a rational number and an irrational number is always rational."

I can easily find examples where the sum of rational and irrational is irrational, as in:

2+√2 : irrational

but unable to find a case where the sum is rational.

Since the statement says "always rational", one single counter-example (as I gave above) invalidates the statement.