Is psychology an art or work? Some are born psychologists and they have passion towards it unlike some people who just work to earn money on it.

The people who just do the same as work, will they satisfy the people with the care which is the prime importance, and is the real burden of talking to the patients concealed?
This is an interesting question, which may or may not garner more than one useful answer. There will be a small element of opinion in the answer which must be very limited for an answer to be a good one for this site, and therefore I will try and formulate my answer with as little opinion as possible.
Your question on whether psychology is an art or work depends much on the differences within your definitions of each.  I will base the difference on the first statement you made:

Some are born psychologists and they have passion towards it unlike some people who just work to earn money on it.

In other words, it is an art if you only need to have a passion for psychology, rather than it being work because you just need to be able to analyse effectively.
This then leads on to the question of whether you can analyse human psychology effectively if you don't have a passion for it or a high interest in it.
@sfxedit cited an interesting study (Gollwitzer &amp; Bargh, 2018), which points out that:

some lay individuals can reliably judge established social psychological phenomena without any experience in social psychology.

with social psychological phenomena meaning

predicting how human beings in general feel, think, and behave in social contexts and situations

You can predict how human beings in general feel, think, and behave in social contexts and situations without actually empathising with the people in question.  A problem with that if you work directly with clients is the fact that clients can detect whether the therapist is empathic or not (Xiao et al. 2015) so can those who are only providing therapy for the money going to be that effective?
That is open to opinion, so I will leave that for you to decide, however, the founder of Person Centered Therapy (Carl Rogers) believes that one of the core conditions for effective therapy is empathy. Plus, from my own personal experience as a client facing therapist, in order for a strong rapport to be built, you need to be able to empathise with your client's situation.

We believe that the most useful definition of empathy would emphasize the ability to detect accurately the emotional information being transmitted by another person. This entity has been termed empathic accuracy by others (see Ickes et al., 1990, for a brief review)   — Levenson &amp; Ruef 1992

So my answer to your question depends on the kind of psychologist you are wanting to be.  If you are a Research Psychologist who is not client facing, then all you need is to be able to analyse the data effectively. Whether the effectiveness will depend on your passion for the subject or not is open to opinion, as is whether the work of a non-client-facing Research Psychologist is work or just an art.
If you are a Clinical Psychologist, or other Clinical Therapist who is client facing, you need to be able to empathise.  For this reason, and the fact that empathising in harrowing client situations can be emotionally draining, it is work in the definition and context put forward. There is however, an art involved in effectively dealing with the emotions internalised through empathy.
So in summary, Psychology and related fields can be seen as an art or hard work.  The decision on that definition is yours.
References
Gollwitzer, A., &amp; Bargh, J. A. (2018). Social psychological skill and its correlates. Social Psychology, 49(2), 88. doi: 10.1027/1864-9335/a000332
Ickes, W., Stinson, L., Bissonnette, V., &amp; Garcia, S. (1990). Naturalistic social cognition: Empathic accuracy in mixed-sex dyads. Journal of personality and social psychology, 59(4), 730. doi: 10.1037/0022-3514.59.4.730
Levenson, R. W., &amp; Ruef, A. M. (1992). Empathy: a physiological substrate. Journal of personality and social psychology, 63(2), 234. doi: 10.1037/0022-3514.63.2.234
Xiao, B., Imel, Z. E., Georgiou, P. G., Atkins, D. C., &amp; Narayanan, S. S. (2015). &quot; Rate My Therapist&quot;: Automated Detection of Empathy in Drug and Alcohol Counseling via Speech and Language Processing. PloS one, 10(12), e0143055. doi: 10.1371/journal.pone.0143055