Let a ring $R$ be defined as $$R=\{ ae+\sum_ {\text{finite}} a_{ij}e_{ij} \mid i\leq j\},$$ where $a_{ij},a$ are real numbers, and $e,e_{ij}\in R\, (i\leq j)$ are linearly independent over the field $\mathbb R$ with $e_{ij}e_{kl}=0\, (j\neq k),e_{ij}e_{jk}=e_{ik}, ee_{ij}=e_{ij}e=e_{ij},e^2=e,$ and the addition in $R$ is in the natural way, and the multiplication is distributive along with the above rules. Is the ring $R$ right perfect?
My try:
If we get $a=a_{ij}=0$ except for $a_{ii}=1$, for all $i$, then we derive from $R$ the infinite orthogonal linearly independent set $\{e_{11}, e_{22},\dots \}$ whose existence is in contrast with $R$ being a right perfect ring. (The set is infinite since $e_{ii}\neq e_{jj}$ via linearly independence of $e_{ij}$'s.)
You're right: a ring with infinitely many pairwise orthogonal idempotents cannot be left or right perfect.
There is a characterization of perfect rings that requires families of mutually orthogonal idempotents to be finite that you can see in Lam's First course in noncommutative rings in the section on perfect rings.