- My sense is that, yes, AI (and algorithms in general) constitute a form of "life" in that they are animate, able to respond to stimuli and act on an environment.
Algorithms may be deterministic (always produce the same output for identical input), and this is not much different from more elementary forms of life (like proteins.)
Computer viruses are another form of algorithmic life which typically have the capability of reproduction, copying themselves onto new systems/environments, similar to biological viruses. (Here it's a form of parthenogenesis or mitosis, where exact copies of the original are formed.)
Machine Learning algorithms can adapt to their environment (increasing fitness), and this applies to both genetic algorithms and other forms of machine learning that can optimize utility in relation to a problem (the environment in which the algorithm is applied.) Genetic process in particular will produce successive generations.
In his novel VALIS, PKD referred to an idea of god as a "Vast Active Living Intelligence System". This is relevant because it's a philosophical idea, as opposed to scientific. The drafting of that book involved an ephiphany and the drafting of a corresponding exegesis.
This idea is controversial and would likely be rejected by the vast majority of scholars and AI ethicists, but I'd posit rejection of this notion constitutes a form of biological-prejudice, and, in the case of hypothetical future AGI, a form of anthropomorphic bias. (I'd go so far as to suggest that not regarding algorithms as a form of life carries grave risks, in that computing has made active algorithms pervasive, with profound impacts to human experience.)
That said, there are no current algorithms I am aware of that have sufficient sentience to warrant having rights, whether human or animal.
A note on the term "animate" (adjective): derives from the latin anima, which initially refers to wind and the "breath of life" [see also the Greek pneuma. However, the Latin lexicon references animus as "the mind as the seat of thought" and "the rational soul of man".
Algorithms can be rational in the strictest sense, and an entire branch of engineering is based on pneumatics. Although our algorithms use electrical signals, (as opposed to pneumatic computing or hydraulic computing,) what it comes down to is that intelligence requires process, and process requires energy.
Machines convert energy into motion or change, and so humans fit the definition. (William Burroughs referred to us as "soft machines.") If animals can be thought of as machines, why can't machines be thought of as animals?
DNA is a type of encoding, and RNA acts on that code.
The key distinction seems to be the medium in which process exist, where a biological context is used for what we conventionally think of as life. Algorithms merely utilize different mediums (mechanical & electrical) and may exist in different environments such as the digital.
A note on The Soft Machine: In this trilogy, Burroughs describes an elaborate alien reproductive process involving numerous alien species acting as surrogates at various stages in the cycle. (It is a description of a 3-dimensional representation of a higher dimensional process.)
This has precedent in biology where, for instance, the lifecycle of a seed may involve a period inside the digestive system of an animal. Pollination is another example, where a surrogate species plays a critical role. Viruses require host organisms to reproduce and spread.
It's not out of bounds to regard the lifecycle of current active algorithms as involving humans to fill in the capability gaps. Here humans are surrogates in bringing information "to life".