With the wording of produce flame, it seems like you take the cast the spell action to make the flame appear. You can then attack with it later – it does use the wording "ranged spell attack." It never addresses what you can do with it otherwise, even though you are still holding a handful of flame.
What if a tavern brawler or a Monk punches with the hand holding the flame, or uses that hand to make a special melee attack (grapple or push?)
Would that be considered a spell attack, or an unarmed strike because the hand is passively holding the flame? Could you use the flame as an improvised weapon?
Or would it be something else?
Note:
I am making a character for a silly one shot. I am not planning on trying to pull this off for a serious campaign, but it is worth trying as a goofy concept. This side note probably won't affect any answers but I wanted to include it anyway.
Best Answer
The flame does not affect an unarmed strike or a grapple
Holding a flame in your hand does not influence the mechanics of an unarmed strike. An unarmed strike is:
Nothing restricts you utilizing an unarmed strike while holding the fire from produce flame, but nothing about the spell description affects the unarmed strike.
It's unclear to me how a hand that is currently holding a flame could be considered free, but even if it is, the same would hold for grapples as would for unarmed strikes. Nothing about the spell implies any different interaction.
Improvised Weapon
In order to qualify as an Improvised Weapon, the thing in question must be an object (emphasis mine):
The fire from produce flame is a spell effect, not an object.
Ask your GM
Since this is for a silly one shot, ask your GM if he can work with you to allow this anyway in some form.
It may be fairly balanced if it is simply changing the damage type of your unarmed strikes or serves as an improvised weapon (dealing 1d4 damage). However, I caution against increasing the numerical damage output of the unarmed strike using this technique. Monks are very capable of dishing out many attacks in a turn, so any increase to the damage compounds quickly.
Additionally, as noted by ThomasMarkov in comments, it takes a 2nd level spell, magic weapon, to do something similar in improving the damage output of a weapon. A cantrip should not be able to do something comparable in power. I recommend sticking with 1d4 as a damage die.