SHORT ANSWER
Unfortunately, we can't be sure, but it is likely that the Carthaginian senate was for life at some point during its long history.
DETAILED ANSWER
The main problem in determining whether or not Carthaginian senators served for life lies in the ancient sources. The qualifications of senators and the precise functions of the senate are also far from clear, and even defining the senate itself is difficult as, in the Punic language,
....there seems not to have existed a word for the 'senate' as a
whole.
Source: The Cambridge Ancient History, vol VI
Punic sources are lost so Aristotle provides the most information (in Politics, which has some of the information from his lost work on the Carthagian constitution) but, as the Cambridge Ancient History (CAH) observes,
His aim and method was to generalize not analyse. This led him to
compare institutions which seem to have little in common, such as the
ephors of Sparta and the Court of the Hundred at Carthage, only
because both exercised essentially a right of control.
Dexter Hoyos, in ‘The Carthaginians’, points to problems with other sources:
Livy once mentions a smaller senatorial body.... Greek writers,
including Polybius and Diodorus, do not help clarity by mentioning at
various times a Carthaginian gerousia (‘body of elders’), synkletos
(‘summoned body’) and synedrion (‘sitting body’), without explaining
the distinctions.
However, the term ‘senate’ is widely used by modern academics writing about Carthage even though it cannot be clearly defined.
When considering the issue of life-membership, we need to look first at the comparisons made by ancient writers. On this, CAH says
It is certain that supreme authority lay with a senate or council,
which Aristotle and Polybius compare with the Spartan gerousia and the
Senate of Rome...
This comparison with the Roman senate and the Spartan gerousia is problematic as these two bodies were very different. However, there are similarities in that members of both the Roman senate (initially) and the Spartan gerousia could be broadly described as aristocratic (for Sparta, in the sense that only Spartiates, a small minority in Laconia, were eligible), and they also served for life. It would seem rash to assume that the same held true for Carthaginian senators as we cannot even be certain
how senators themselves were recruited, or even how many there were at
any time, although two or even three hundred is likely
(source: Dexter Hoyos, ‘The Carthaginians’)
That said, if Carthaginian senators were from privileged backgrounds (which seems likely based on the available evidence), it would not be unreasonable to state that lifetime membership was more likely than not. The Oxford Dictionary of the Classical World goes so far as to state
There was a powerful ‘senate’ of several hundred life-members
but there are few other modern sources which make such a definite statement (here is one).
Finally, we should also not forget that there were several political upheavals in Carthage over the centuries and we certainly cannot assume that life-membership of the senate or the form and power of the senate remained unchanged during all that time.
Other sources:
Richard Miles, 'Carthage Must Be Destroyed'
M. I. Finley, 'Politics in the Ancient World
David Abulafia, 'The Great Sea'
https://en.wikipedia.org/wiki/Ancient_Carthage#Government
http://www.roman-empire.net/republic/carthage.html
http://www.crystalinks.com/carthage.html
http://www.newworldencyclopedia.org/entry/Carthage
