Roma, who also received two red cards in the space of two minutes, were beaten 4-2 after extra time.
When asked about the error, Fonseca told reporters: "If there was a problem, that I honestly didn't know about, we'll discuss it internally."
Italian FA rules state that teams can make five subs in three slots through the 90 minutes. Should a match go to extra time, teams have an extra slot so long as they have not already used their five subs.
Spezia had raced into a two-goal lead in the capital through an Andrey Galabinov penalty and Riccardo Saponara after 15 minutes.
But within two minutes of the restart, Roma were down to nine men. First, defender Gianluca Mancini received a second yellow card for a foul just inside his own half and, from the resulting free-kick, goalkeeper Pau Lopez tried to clear the ball, but completely missed his kick and took out Spezia's Roberto Piccoli instead and was dismissed.
Daniele Verdi's header put Spezia back in front in 107 minutes, but a stunning Saponara chip one minute from time confirmed the newly promoted side's place in the quarterfinals.
Information from Reuters was used in this report