Immigration and tenure track in the United States
I've taught at both large R1 universities and small liberal arts colleges. For the large R1s, we don't care -- we simply want the best person for the job, regardless of their nationality. R1s have very competent foreign scholar offices and lawyers who know how to do the proper visa paperwork.
At smaller schools, it's more difficult. Budgets are tighter and the provost has more say over searches. If there are two candidates that are almost equally ranked, then the provost may suggest one over the other for non-academic reasons. There may not even be approval for international airfare for job candidates at the interview stage.
You have to remember that an H1B application will cost at least $10,00-$20,000 in lawyer's fees, staff time, and application costs. The stack of paperwork for the Department of Labor certification itself is about 3 inches thick. Furthermore, even though there is an exemption, sometimes H1B applications can be very, very slow -- which means that an instructor hired in March might not be able to join the campus by August. This can be deadly for a small school that needs its faculty present.
tl;dr Be the absolute best candidate possible for the position and the visa status won't matter. Otherwise, it is a factor amongst other factors.
This question is about visa issues, which aside from being a legal matter are notoriously fraught with subtle technicalities, to the extent that even a lawyer may need many very specific details in order to answer a question on the subject. So let me start by saying that I am not a lawyer, nor a person with expertise on immigration matters, nor someone agreeing to be held responsible for any advice I am offering here. I am just a friendly person with an opinion, which I am happy to offer. I am also currently a department chair at a large U.S. university, so have a bit of experience dealing with visa-related questions on the hiring side.
The main points that I think are relevant for your question are the following:
As Nate Eldredge commented, U.S. universities are not subject to the quota system for H-1B visas. When a university wants to hire a foreigner, they can sponsor them for an H-1B visa. Assuming you are qualified, and assuming that the NSA doesn't know anything bad about you that we don't, I don't see a reason why it should be even remotely a problem for you to obtain a working visa for the U.S.
It is true that sponsoring a new faculty member for a visa costs money. I don't know exactly how much; the $20K figure mentioned by RoboKaren sounds too high to me, but for argument's sake let's assume that it is correct. Well, although it is objectively not a small amount of money, it is in fact very small if you compute it as a percentage of the cost of employing a faculty member. At a good research university, a new faculty member in CS (not my area, but close enough for me to know something about) will typically be offered a startup package of at least $50-100K, and possibly quite a bit more, to pay for startup costs such as relocation, buying some equipment (computers, and other things depending on the kind of work the faculty member does), and to allow paying for travel to conferences, hiring research assistants and such for a year or two until the person is able to secure some grant funding. And then there is the cost of the faculty member's salary, which will be paid indefinitely until he or she retires.
The point is that when you consider the hypothetical $20K expenditure in the context of the total benefit a good researcher brings to the institution, and the total cost of their employment, the $20K is actually a trivial amount of money. For a serious university, even a poor one, I think it is a no-brainer that it is worth making the effort to hire the best possible person for the job, even if that requires a modest extra investment of that sort, compared to hiring a U.S. citizen for whom that expense is not needed but who is slightly less qualified. I have never heard of a department that made the opposite calculation, and I have never heard of a foreign researcher at the tenure-track level who experienced difficulty being hired because of immigration issues. That does not mean such things don't happen, but if they do then that would be outside the realm of my own experiences.
Lastly, you mention that you have never visited the United States. Although you asked specifically about visa issues, and my opinion is that visa issues are mostly or entirely irrelevant to your situation, I think it's worth mentioning that the fact that you've never visited the U.S. is in my opinion (very) relevant to your situation, and is a much bigger complication and potential hindrance to your job applications than anything related to visas. It's not that I think there's anything wrong with not visiting the U.S. -- far from it -- but I do think your lack of familiarity with the U.S. and its culture could be a real problem both when you come for an interview and if and when you move here after getting a position. The very fact of your applying for a high-level position in a country you have never visited could appear quite strange to people considering your applications, leading them to question how well you are likely to fit in at their department and institution. I don't know how much thought you've given to this issue, but I would strongly suggest that if you have a serious goal of securing a tenure-track position in the U.S., if at all possible one of the first steps in your plan should be to arrange to first of all visit the U.S. for a reasonably long period of time, say a month or two, and visit at least a couple of different geographical parts of the country, and certainly include some universities in your visit so that you can experience first-hand what academic life in the U.S. is like.
Note that I absolutely don't mean for this part of my answer to be read as in any way a criticism of your desire to move to the U.S. or of your never having visited. I am merely saying that coming for an extended visit first will (in my opinion) make you a much better job prospect and a much better interviewee, in a way that I think you cannot appreciate given your lack of U.S. experience. By the way, although this opinion I'm expressing is specific to the U.S. and is not entirely symmetric with respect to countries, it would definitely apply in many other analogous situations; for example, I have never been to Brazil, and if I were to apply for a tenure-track position in Brazil without visiting first, my guess is that my lack of Brazilian experience (separately from my lack of knowledge of Portuguese) could be a serious disadvantage for my applications. (Though, of course, never having been to Brazil, I cannot be sure if that is indeed the case... :-) )
Caveat: I'm a US citizen, so my experience with this is from the hiring side and from discussions with colleagues who are not US citizens.
The short answer is that it will not change things very much. All of the hiring processes I've been involved with paid absolutely no attention to a candidate's citizenship status. That doesn't guarantee this is true everywhere, but in most cases, I don't think it's a big issue. Universities have it much easier than almost any other employer under US immigration law (for example, they have an exemption to the H-1B cap), and while there will be a fair amount of legal runaround, I've never personally known someone hired to a TT job in the US who wasn't able to take the job for immigration reasons.