In this context, I think two words would work.
Instigator - a ​person who ​causes
something to ​happen, ​especially something
​bad or Inflammatory person
As we use 'inflammatory remarks' to denote something that causes something bad, inflammatory person is the one who causes dispute by adding fuel to the fire
ell.stackexchange.com/users/28523/Blahmastah
Best suggestion: conflict escalator .
Also acceptable: argument escalator.
The term for increasing an argument would be to "escalate" the argument.
Unfortunately the term "escalator" would normally be held to mean an actual "moving stairs" escalator, such as in a building with multiple floors, so it is awkward to try to describe this concept with a single word.
If an argument, or disagreement, between Amit and Shiva starts out with polite words, but then Amit begins insulting and cursing Shiva, one could say "Amit is the one who escalated the argument".
If you said "Amit was the escalator" it would most likely draw puzzled looks, as people try to figure out how Amit used to be moving stairs made of metal, but somewhere along the line changed into human form.
It would sound appropriate to label someone as a "conflict escalator".
For example, one might say, "His attorney is a conflict escalator" to describe an attorney who looks to instigate fights between the parties involved.
(Here, instigate means to start an argument, where there wasn't one before.)
The other available words mentioned thus far, catalyst, and instigator, are a bit too general to capture the sense of a back-and-forth argument.
A catalyst would be someone whose presence is required for the argument to occur, but does not necessarily participate in the argument, such as two men arguing over a whether the bartender's shirt is blue or purple.
The bartender (their shirt, to be precise) is the catalyst, but may not be involved in the argument, or even aware of it.
An instigator would be the person who starts , i.e., instigates the argument, such as by verbally attacking the other person.
Opinions can differ on who really started the argument, but the term instigator means the one who started it.
The term "inflammatory person" is a general negative label of another person.
It is not a specific description of the behavior of escalating an argument.
The one who "escalates" the argument would normally refer to a person or party who is directly involved in the argument, rather than an outside agent like a catalyst.
ell.stackexchange.com/users/28526/Developer63
Agitator comes close and could work, although this is not the normal context for it... ...as well as incendiary , although, again, this is not commonly used in your context.
ell.stackexchange.com/users/27803/GoDucks
Stirrer is a British usage.
Depending on your audience perhaps trouble stirrer or argument stirrer would be clear enough and also appropriate.
Troublemaker might fit as well although that can encompass as lot of things.
ell.stackexchange.com/users/8026/shawnt00
Believe it or not, there is a Positive way to fuel the fire : DEVIL'S ADVOCATE noun 1.a person who expresses a contentious opinion in order to provoke debate or test the strength of the opposing arguments An alternative definition, and more appropriate to a friendly debate is: Someone who takes the other side of the argument purely for the sakeof argument .
The one who stokes the flames of an argument in a less positive light is an Antagonist : ANTAGONIST noun 1.a person who actively opposes or is hostile to someone or something;
an adversary Some good synonyms are: obstreperous truculent vociferous
ell.stackexchange.com/users/26773/lurker
For figurative speaking occasions: accelerant A substance that accelerates the development of a fire. https://en.wiktionary.org/wiki/accelerant https://en.wikipedia.org/wiki/Fire_accelerant
ell.stackexchange.com/users/28309/lauir
Merriam-Webster says one who provokes is a provocateur .
ell.stackexchange.com/users/55794/apocalysque
What can I name a person who increases the argument between two people.   Can I call him as  catalyst  or is any other word available?
Word for a person who increases the argument
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<word-request>
ell.stackexchange.com/questions/78011
ell.stackexchange.com/users/28470/Arulkumar
Stirrer, troublemaker, devil's advocate, antagonist, provocateur, and  agitator are all terms that may be used for a person who provokes an argument. Stirrer, troublemaker, devil's advoate, antagonist, adversary, accelerant, provocateur, escalator and agitator are all options.
Stirrer, troublemaker, devil's advoate, antagonist, adversary, accelerant, provocateur, escalator and agitator are all options.