In a book on postal culture in early modern Europe, the author mentions how the Tassis family “developed an extensive network of post roads...” in the fifteenth and sixteenth centuries. What resources would a family need to construct such roads? Who would they hire?

It would be more accurate to speak of the creation of postal routes rather than roads. In fact, one of the key reasons an early Reichspost was even possible was the Taxis'--that's the German! :) --use of existing infrastructure. Roads (or more frequently "roads"; think more of a roadway than a via romana for most of the distance), inns, stables, banks...the Taxis' success lay in the creation of a network of what you might call subcontractors, initially under Maximilian at the end of the 15th century and more permanently starting with Charles V a few decades later.

The Tassis or Taxis family, eventually raised to the German nobility as "von Taxis," was well-established as a courier company when Max hired them to develop and operate standing, ready-to-go, cost-effective postal routes between his location and key points in his empire. Their experience as and managing couriers meant the Taxis brothers well understood the biggest limiting factor in the age of horses: biology. Travel was hard on humans, and harder on horses.

As Marjorie Rivers reconstructed, a typical "day's journey" in the late Middle Ages was somewhere around 30-33 miles per day, usually assuming one person mounted with at least one member of the party traveling on foot. A single mounted rider, she found, could reach up to 50 or even 55 miles on one day. But this pace was (a) unsustainable for the human (b) absolutely not feasible with one horse the entire way (or even sometimes on the same day). The rider would need to switch out mounts along the way. The problem was, that would mean the ability to locate and rent/buy a new horse exactly when they needed.

The Taxis built a network to cover that "when they needed."

They made deals with innkeepers and stablemasters (often the same people) at set staging points along key routes. In exchange for a flat and fixed fee, the "subcontractor" would have horses and usually riders standing by at all times to carry imperial-business mail. To fund this endeavour, the Taxis had their own payment from Maximilian's treasury but also borrowed from the Fugger merchant-banking family. (Who were probably also the source of Maximilian's payments, in the end...)

More importantly for their immediate financial profit and also the long-term, lasting success of a Reichspost, was their decision to involve banks/banking families along the designated postal routes in the process. The banks, which had plenty of long-distance business to conduct, could buy into the otherwise government-only (which was not just the imperial court, remember) mail system. This was a precursor to the eventual idea of access to everyone with the money to send messages.

...Oh, yeah, and the Taxis also made money opening letters and selling the contents as intelligence. "The postmaster, and therefore also the spymaster" indeed.