Romance?
Personally, I believe there is potential for romantic feelings between Jessie and James. Even disregarding their married-couple-esque fighting, they do have actual emotional and mental bases for becoming a couple, using Robert Sternberg's triangular theory of love, which states, in brief:
mental & emotional intimacy
+ commitment
+ physical passion
= ideal love
Mental and Emotional Intimacy?
My vote is yes. Jessie and James apparently like each other well enough to travel together for extended periods of time, and as we see many times in the Indigo League episodes, they know each other's moves almost before the other person performs them. They also implicitly trust each other, as their classic "Team Rocket" back-to-back stance shows. They have had to get to know each other very well if they're going to work together as part of Team Rocket, and the fact that they haven't split up permanently despite many failures is a sign of knowing and appreciating each other well.
Commitment?
My vote is yes. First brought together by Giovanni, staying together due to their commitment to Team Rocket and its ideals, Jessie and James seem to be just as committed to each other as to Team Rocket as an organization. Neither one looks for ways out of the partnership, even when their schemes fail so miserably, so there's something besides duty holding them together, otherwise they would have applied for different partners by now. (Example: in the above image, Jessie has just lost the Princess competition and her chance at a set of Pokemon dolls she never had as a child. James, with Meowth's help, organizes a sort of consolation prize--dressing up as the Pokemon dolls, to remind her she still has their support.)
Physical Passion?
My vote is...maybe, given time. Sure, the anime doesn't overtly show them being sexually attracted to each other (given that it's more geared toward kids), but even so, there are a few important subtexts, not least the above delicious scene at the end of "Holy Matrimony" with James and Jessie floating away into the sunset together in the Team Rocket balloon:
Jessie: (to Meowth, as they walk together) Oh well, guess James chose to hang up his Team Rocket costume.
Meowth: Ehh, I guess all that money and luxury was just too tempting for James to give up.
Jessie: (sighs) I guess you're right.
James: (from above, in the Team Rocket balloon) Prepared for trouble?
Meowth: (looks up) Hey, it's him!
Jessie: ...James? (She smiles) Make it double! (She laughs)(James maneuvers the balloon closer to the ground; Jessie runs to catch up with it, and James catches her by the hand and pulls her up into the basket)
Jessie: I guess we're not going to get rich this time, either.
James: Guess not. They wouldn't give me the inheritance.
Jessie: Oh, there's always next time. We may not make a lot of money, but we sure have got our freedom!
James: Yeah. Double trouble time, right?
Jessie: Sounds great to me, James.
(They clasp hands, exchange a meaningful look)
(cue 13-year-old me squealing and bouncing on the bed)
It Doesn't Have to Be Romance, Though!
The mental and emotional intimacy that Jessie and James have, combined with their apparent commitment to each other, could indeed breed a physical attraction in time. Best friends sometimes do become lovers--and spouses! BUT it could become instead a rock-solid friendship in which both work together and cheer each other on to find love. This would be just as legitimate a development. (Source: personal experience.)
As a 13-year-old first watching Pokémon, I was admittedly a rabid Rocketshipper; as a woman in my early thirties, I still find a lot to love about Rocketshipping, though I'm quite a bit tamer now. For me, something about the pairing works--perhaps it's the "opposites attract" aspect, or the beauty in how their strengths and weaknesses play off each other. Or perhaps it's the idea of two wounded people finding each other and healing together. But I fully support others seeing this duo differently. That's the great thing about implied elements of fiction: no single fan interpretation has to be correct. Writers like me just have fun seeing what people find in our stories!