You should keep in mind that the people running searches are busy academics who are doing this as service to the department. Checking-in aggressively — following your intuition for how this would work in business — will annoy and will be very unlikely to help. At the moment (early/mid- February), it is still early enough in some job markets that interviews have not been scheduled so asking for updates might be seen as pushy. I think it's unlikely that your chances will go down if you ask, but it still might be nicer to wait.
That said, there are many situations where contacting in normal and you might to do it through one of those channels:
For example, if you have another offer from somewhere with a deadline, it is normal — and a good idea — to contact other departments to let them know that you will have to move forward without them. I've had friends who have had interviews offers within hours of telling a department this.
Also, if you have updated material on your CV (e.g., a paper accepted, positions changed, an award, whatever) go ahead and send your updated CV. You can mention in that email that you're excited to hear about any updates from their search.
Contacting search committees in this context is normal and can signal that you continue to be very interested in a job. I was told by a search committee that ended up making an offer that they thought I might be unlikely to accept an offer and that one of these update emails rovided a useful signal. Of course, if they're not interested, emailing will probably just be noise and extra work for them.
I think that emailing or not-emailing is unlikely to tip the scale either way. They're going to make a decision based on the intersection of the quality of your work and what they're interested in having in their department. But out of kindness for the work of the search committee and its chair, try to do it as little as possible.