Spurs were eventually beaten 3-2 thanks to Antoine Semenyo's emotional 95th-minute winner ahead of his move to Manchester City.
Post-match, Frank was asked about the Arsenal coffee cup, which he insists he had no knowledge of at the time.
"I definitely did not notice it," said Frank afterwards. "It would be completely stupid of me to take it if I knew.
"It's a little bit sad in football that I need to be asked about it. I would never do something that stupid.
"I think we're definitely going in the wrong direction if we need to worry about me having a cup with a logo of another club."
Meanwhile, BBC Sport is reporting Frank was handed his pre-match coffee on his arrival at Dean Court by a Spurs staff member. The cup used was a leftover from the cups brought by Arsenal, who played at Bournemouth only four days earlier.
The staff member nor Frank noticed the Arsenal crest until fans were spreading images across online.
On the defeat, Frank added: "We are all hurt, the fans are hurt. That's absolutely natural. That's why football is emotional. It's just very painful for everyone involved with Tottenham."