What would be the professional consequences for a theologian if he denied publicly the existence of Jesus Christ?

The professional consequence is 100% dependent on the terms on which the theologian is hired to do. Various institutions have different purpose for a position that employs a theologian.

If the position includes the responsibility to teach, defend, and represent the faith statement of the institution (church, school, denomination, academic society, magazine, etc.) AND if the faith statement includes the belief that Jesus Christ is a historical fact, then the theologian needs to do his/her job in a way consistent with the faith statement. Of course, not every item in the statement is of equal importance, but historical existence of Jesus is absolute core, as @curiousdannii puts it:

I'm not talking about a rational and healthy scientific skepticism over narrow details in a field, but a complete ideological shift to completely opposing the entire enterprise of your employer.

What if the theologian personally doesn't agree with Jesus Christ as a historical fact? Then he/she has the option to keep it private while doing the duty. But if he/she cannot keep the cognitive dissonance and feels compelled to declare publicly his/her objection, the institution can then legitimately declares a conflict of interest or a scandal, and proceed to propose a resolution that results in forced resignation / termination.

Many terminations or voluntary resignations have happened for a lot less than this core belief of most Christian groups, so it is no surprise that this theologian should expect to resign / terminated if he/she goes public:

• Evangelical Theological Society president Francis Beckwith resigned his post when he came back to the church of his baptism, the Roman Catholic Church
• Peter Enns, a tenured professor at Westminster Theological Seminary, was suspended over mere controversy caused by his book that is perceived to be in conflict with the Westminster Confession of Faith.
• Hans Küng was not allowed to teach as a Catholic theologian over his objection to the doctrine of papal infallibility, but he remained a Catholic priest until his death and his professional position at Tübingen was not impacted.

On the other hand, if the position does NOT have the purpose to defend Jesus as historical fact, for example in many secular universities's religious studies department where research and publications is the #1 benchmark, then they probably don't mind you declare this publicly. As @MikeBorden points out, Bart Ehrman is flourishing professionally in the department of Religious Studies at University of North Carolina at Chapel Hill.

With all that said, practically no serious scholars today deny the existence of Jesus. Even Bart Ehrman believes that Jesus existed and was crucified. It's his divinity and his resurrection from the dead that are at issue.

Responding to comment

I thought, who are the researchers studying those topics mostly and lay the foundations about the historicity of Jesus Christ. I guess that those are theologians and historians that are also theologians. For their current and or future career, they are therefore not able to freely research this topic, but their outcome is predetermined. So I was always interested what the ratio between researchers with a theological background and researchers without it and their respective outcome about the historicity of Jesus Christ.

I don't know and don't care about the existence of a historical Jesus Christ. I just had the feeling that the research is strongly biased, and I don't think that some exceptions to this assumption help here. But I know a little about Ehrman. But it helped me to see some of the possible influences on researchers in this area.

Yes, I agree with you that bias can run very deep in these institutions. I read several theologians who purposefully stay quiet to survive and only later after retirement feel safe enough to go public. But these tend to be in grey areas that are not as core to the faith, such as Open Theism, tweaking Chalcedon, going against the official decree of studying Thomism, not accepting Chicago statement of inerrancy, etc.

There is enough diversity TODAY that various Christian denominations' institutions have freedom to publish their own positions. So as a theologian you have the option to go to an institution that supports your favorite flavor of Christianity. But if you need to earn money and the viable option is not ideal for you, you have to deal with certain cognitive dissonance until you have well enough footing financially to change a place of employment. Do the research result got compromised? I don't think so since reputable Christian research institutions have a peer review process, and that the scholarship stands on its own merit open to review by competing parties (such as Catholic vs. evangelicals). So most of Christian academia, just like the secular counterparts, has its own corrective mechanism.

I think you need to make a distinction between core beliefs vs. gray area where Christian groups don't agree. Core beliefs have been debated over 2000 years. Prime example is the Trinity, that is still debated even today. Don't let false narratives fool you, thinking that the orthodox definition of the Trinity either came from above (by authority, i.e. Constantine forcing the bishops to meet) or by sociological pressure (i.e. orthodox bishops man-handle non-orthodox ones). Reputable church historians would assert that there were enough freedom back then as well as today for every variation under the sun to get a hearing. The question for historical variation is whether they survived the ravages of time, but FEAR NOT, it can be rehashed today!

As for your example, the historical existence of Jesus, virtually all scholars of antiquity accepts this based on external evidence OUTSIDE the church. See Wikipedia article section Historicity of Jesus : Non-Christian sources for typical primary sources that secular historians use.

Conclusion

Given

• the thousands of Biblical scholars and theological researchers today in hundreds of Christian academic institutions
• the overall climate / ethos of being ecumenism-minded
• the propensity for research-level Christian institutions to model secular academies
• that personal blogs of theologians are flourishing

there is enough freedom in the Christian academia overall, unlike in totalitarian states like China, to ensure that the outcome can stand on its own. The test of time IS another controlling factor, where "worthy" positions are kept being referenced in the bibliographies and grew more popular, some became "great" (like Karl Barth's magnum opus).

The bias will INSTEAD come more from the consumers who select which theological "products" to believe from the marketplace of books and journals which (for now) is relatively diverse and free.