The choice between [to do] not have to and [to be] not having to is a stylistic choice, but one which accentuates the “right now” quality of a continuous state through time. In this instance, the present continuous makes the present more present to the reader. Using a simple tense merely makes a statement of fact.
people now travel with BlackBerry and with mobile phones, so they're not actually having to use hotel telephone services.
That people now travel with mobile devices is couched in the simple present as a fact: wind blows, rain falls.
The result in a continuous form suggests a series of guests, all equipped with iPhones, demanding free Wifi.
The same effect could be obtained by using a continuous form for the first verb and simple for the second:
People now are travelling with BlackBerry and with mobile phones, so they don’t actually have to use hotel telephone service.
Using continuous for both verbs would be overkill. Once the continuous aspect is introduced, it need not be again.
But there are other details you’ll be free to observe if you aren’t having to keep your eyes glued to a notepad. — Cynthia MacGregor, When I Grow Up, I Want to Be a Writer, 2001, 36.
Fortunately, there is a cease-fire agreement now so we aren't having to cope with additional war-inflicted injuries and casualties. — Azerbaijan International 3(1995), 54.
Finally and most importantly, you want your customers to do a lot of your advertising for you. Think about ways you can engage your audience so that you aren't having to do all the work yourself. — “Eight questions to ask before launching your social media marketing campaign,” 24 May 2014, Aviaro.com
Changing your air filters once every three months during the cooling season will ensure that your air conditioner isn’t having to work too hard to cool your home. — Morgantown Air Conditioning and Heating
You can change the verb in each of these sentences to doesn’t/don’t have to without serious loss, but the continuous gives the described state a certain immediacy. In the last sentence, for instance, I appreciate the fact that changing the air filter makes an air conditioner more efficient. But with the continuous, I’m in a poorly cooled room listening to the labored sound of an AC with a dirty one clogged with pet hair and dust.