The answer is yes, as there is no indication that SARS-COV-2 is an exception to the rule.
Cp., for instance: SARS-CoV-2 triggers inflammatory responses and cell death through caspase-8 activation, Shufen Li, Yulan Zhang, Zhenqiong Guan, Huiling Li, Meidi Ye, Xi Ch https://www.nature.com/articles/s41392-020-00334-0
On exceptions to the rule cp., for instance
V. Kaminskyy, B. Zhivotovsky, To kill or be killed: how viruses interact with the cell death machinery
https://pubmed.ncbi.nlm.nih.gov/20433575/
In my opinion, only in the context you cited your question seems a still open one:
https://www.sciencemag.org/news/2020/12/coronavirus-may-sometimes-slip-its-genetic-material-human-chromosomes-what-does-mean
"... Baltimore says. “It is also not clear if, in people, the cells that harbor the reverse transcripts stay around for a long time or they die.”
However, this statement does not contradict the general rule that cells infected by viruses finally die.
There is always, as cited, some "staying around" - otherwise the immunogenic mechanism of presenting antigen would not exist.
As for retro viruses, host cells might divide, thereby multiplying the integrated genome of the virus, so one might speak of "no dying" as in such case the cell is still able to divide. However, that should be seen as only "staying around" for a very much longer time, as every infected cell will sooner or later become the retro viruses' friendly host - if that cell hasn't died before because of other reasons:-) Still, if CoV-2 were a retro virus the answer to question had to be "no" - but it is not, CoV is no retro virus.
Refering to integration by LINE elements: this is different from retro viruses and it might be an open question, see the text passage cited, if the integration as a coincidental and irregular one (that should be current common belief) does interfere with viral replication of CoV-19. In the study cited it is suggested that the RNA of the virus that has been retrotranscribed and integrated into the host's DNA is being read and transcribed to viral RNA. Interestingly, at that point the study speaks of viral "chimeric RNA" being detected, not viral proteins that maybe have not been searched for. The study also speaks of antigen that has been transcribed and translated from retro-integrated RNA may still be presented (on MHC) to immune cells. The latter possibility might be ruled out if viral proteins or peptides have not been found - in that case it is much more probable that the expression of viral genes has stopped the host from producing virions which is in line with a known strategy of "restriction" (e.g. by small or large interfering RNA).
In other words: to me it is highly probable that the reading of integrated viral RNA is a mechanism of "restriction" in a wider sense in that by activating the transposon after integration the cell activities that would help the invading virus to replicate are being halted which different from o known defence mechanisms in bacteria/procaryotes leads to cell death probably by apoptosis, but maybe by necrosis.
This indeed seems coherent to "cytokine storm" as a reaction to SARS-Covid not of the adaptive but the innate immune systems that tries to clear up dead cells.
Another argument pro cell death is the fact that - maybe contrary to common belief - it does not seem a seldom and random event that viral RNS becomes integrated as the study's outcome apparently is a "first hit - success". To my mind, the - disputable - fact that there are no viruses that infect cells of the germ-line with the exception of certain retro(!)-viruses, cp. https://en.wikipedia.org/wiki/Endogenous_retrovirus, https://www.sciencedirect.com/science/article/pii/S0014579300016069, https://www.sciencedaily.com/releases/2019/10/191010113231.htm can be reconciled with and thus corroborates my opinion that integration of viral genes as transposons a. is no rare event and b. leads to cell death in eucaryotes because, otherwise, the integration of transposons the activation of which leads to apoptotic processes in stem cells might put the indivual if not the host species at risk.
In my opinion it is a possibility that the LINE insertion is part of a yet rather unknown "restriction process" of certain infected cells that interferes with viral replication. The study found viral RNA transcribed from DNA that had been successfully integrated into cell's genome. The RNA found in samples might be some "interfering RNA" of some mechanism of "restriction". If this is correct, it seems a possibility that cells stays alive with viral genes integrated and - refering to maybe the gist of your question - divide and multiply, in an attempt to replace other cells who were not successful with this restriction strategy. Moreover, those successful restrictive cells might (seems like a paradox) NOT present antigen to cytotoxic T-cells, and not get killed by those. However, I think this is highly unlikely. Much more coherent with findings of chimeric RNA is some induction of cell death by expression of integrated viral genes. It is known that active transposons can interact with genes they transpose to, leading to heritable disease (in eukaryotes). That should be - next to adaptive immune system defence - another way infected cells die altruistically, in order to stop the progression of viral invasion.
In any case, the answer is: yes, cells infected by SARS-CoV-2 die.
Still, it is a possibility that "successful restriction" (integrated RNA becomes interfering RNA) leads to destruction of the virus, as this seems to be a known defence outcome in the bacterial world. However, the findings - RNA detected outside cells - suggest that apoptosis or necrosis is the way to die for human cells infected by CoV-19.
Hint: In any case, the fact that killer cells of the immune system lead to apoptosis of infected cells does not imply that a cell would survive an infection (if it were not for the killing by killer cells).
