Dean brought up Bert and Ernie three times during the run of Supernatural.
5x03 - Free to be You and Me
The first time was when he and Sam were broken up at the beginning of season 5 and he decided that he was going to find Castiel a woman to have sex with, so he takes him to a brothel. This was presented as a sort of "Bert and Ernie may be gay but we aren't so let's go pay a woman to have sex with you" type thing.
Dean: Well. Last night on earth. What are your plans?
Castiel: I just thought I'd sit here quietly.
Dean: Come on, anything? Booze, women... You have been with women before. Right? Or an angel, at least? You mean to tell me you've never been up there doing a little cloud-seeding
Castiel: Look, I've never had occasion, okay?
Dean: All right. Let me tell you something. There are two things I know for certain. One, Bert and Ernie are gay. Two, you are not gonna die a virgin. Not on my watch. Let's go.
11x08 - Just My Imagination
The second time was when he and Sam dressed up as trauma counselors, in cardigans. Now, since he had previously stated that he thought Bert and Ernie were gay, this implies that he felt that he and Sam looked gay in the outfits. Sam implies that Dean insisted on coming along with him.
Dean: Good. The Bert and Ernie pretext. Awesome.
Sam: Dude, you didn’t have to come.
The third, and final, time was when Dean suggested that he and Sam dress up as the Muppet pair as a couple's costume idea. The fact that he immediately said that it was a weird suggestion indicates that he still thinks Bert and Ernie are gay and realized that it might not be an appropriate choice for him and his brother. But then he goes on to suggest Moose and Squirrel (hehe) and then two human and dog combos and then a cat and dog combo, all bizarre but funny. Then he suggests Thelma and Louise and double downs on that one. He was not being serious about any of these, he was being silly to try and lighten the mood.
Dean: Alright this is what we’re going to do, next year we’re doing Halloween right okay? I’m thinking matching outfits like uh, Batman and Robin.
Sam: No.
Dean: Bert and Ernie.
Sam: No.
Dean: That’s weird.
Sam: Yes.
Dean: Uh, Rocky Bullwinkle.
Sam: Dean…
Dean: Shaggy and Scooby!
Sam: Why would we?
Dean: Turner and Hooch. Ren and Stimpy.
Sam: Come on now.
Dean: Thelma and Louise.
Sam: Thelma and…? Dean: We just put it in drive and go.
Dean also mentioned Thelma and Louise to Castiel in 5x03 when Cas was talking about his plan to trap and interrogate Raphael. Castiel, of course, did not get the reference, so did not know that Dean was saying that the plan was suicide., so Dean changed to a more direct, less pop culture dependent approach.
Dean: So, what, I'm Thelma and you're Louise and we're just going to hold hands and sail off this cliff together?
So, basically, he referenced Bert and Ernie as counterpoint to him and Castiel (they might be gay but we aren't), compared himself and Sam to Bert and Ernie (we look gay), and then suggested he and Sam be Bert and Ernie... but realized that would be weird. And then suggested he and Sam be Thelma and Louise, which he'd previously equated to suicide, because Sam is his ride or die.
This isn't about ships because the show wasn't about ships. But what the show was about was Sam and Dean.
