Language Transfer is the term for influence of the mother tongue (L1) on the production of the target language (L2). Language transfer can be both positive and negative. So, for example, positive transfer accounts for L1 German speakers saying something like:
Careful! The glass has a crack! (Vorsicht! Das Glas hat einen Sprung!)
since the use of the definite and indefinite article is identical in such a context. But the same German speaker might ask:
Do you play piano?
since in German no article is used (Spielst du Klavier?), whereas the definite article is needed in English (negative transfer).
It would seem therefore that there is a clear and inevitable influence of the L1 on the production of L2. Indeed, this was the strong position taken by early researchers in Contrastive Analysis. For example, Lado in his Linguistics Across Cultures (1957) hypothesized that:
those elements which are similar to [the learner's] native language
will be simple for him, and those elements that are different will be
difficult.
But as Odlin notes in the introduction to his book Language Transfer: Cross-linguistic influence in language learning (Cambridge University Press, 1989) the increasing amount of empirical data about L2 learner errors that was produced in the 1970s called Lado's hypothesis into question. Odlin gives the example of the common omission of the copula by L1 Spanish speakers of English:
... speakers of Spanish, which like English, has copula verb forms,
frequently omit forms such as am or is. Moreover, such errors are not
only found among Russian and Spanish speakers, but also among speakers
of other languages - and also among children learning English as their
native language. Thus, while a contrastive analysis might explain a
Russian speaker's omission of copula forms, a Spanish contrastive
analysis would not explain the same error as they acquire English. The
pervasiveness of certain types of errors has thus been among the most
significant counterarguments against the importance of transfer.
In the rest of his book Odlin analyses these counterarguments, but in the end comes to the conclusion:
Despite the counterarguments, however, there is a large and growing
body of research that indicates that transfer is indeed a very
important factor in second language acquisition.
Learner English: A teacher's guide to interference and other problems (ed. Swan & Smith, Cambridge University Press, 2001) aims "to help teachers anticipate the characteristic difficulties of learners of English who speak other mother tongues, and to understand how these difficulties arise".
The book is clearly based on the conclusion that Odlin came to, namely that L1 transfer plays a very significant (but not exclusive) role in the production of L2. The introduction has the following text, with various references to article use:
There is less disagreement than there used to be about how far
interlanguages are influenced by learners' native language, and most
linguists would probably now agree that the mother tongue can affect
learners' English in several ways:
Where the mother tongue has no close equivalent for a feature,
learners are likely to have particular problems in the relevant area.
Japanese or Russian students, for example, whose languages have no
articles, have a great deal of difficulty with English articles.
Where the mother tongue does have an equivalent feature, learning English is
in general facilitated. French or German-speaking students, for
instance, find articles relatively easy in most respects, despite the
complexity of the system.
However, equivalences are rarely exact, and
so-called interference or transfer mistakes are common where students
assume a more complete correspondence than exists, so that they carry
over mother-tongue patterns in cases where English forms or uses
are not in fact parallel. French or German-speaking students typically
make certain mistakes in English (e.g. *The life is hard, *My sister
is hairdresser) precisely because their languages do have article
systems.
So, finally, the answer to the question Does the fact that a learner's native language doesn't have articles affect their ability to use them in an L2 with articles? is an emphatic Yes! as far as English is concerned.
I see this every day with my English learners. Those with Japanese, Chinese, Korean or Russian L1s make significantly more mistakes than those with German, French or Dutch L1s. And the learning of the article system seems all but impervious to direct instruction. What works best is copious exposure to written and spoken English, coupled with an ability and willingness to notice article use in natural contexts.
Wikipedia links:
Language transfer
Contrastive analysis
Interlanguage
Noticing