According to Sydney Griffith's The Church in the Shadow of the Mosque, Oriental Christian Churches (and I do not use the word in a pejorative sense, it is the term of choice for both practitioners and scholars of the various Eastern Christian sects) from the moment of the Islamic conquest have been under gradual but consistent decline. On occasion this decline has accelerated for brief periods of time, but it has never reversed. Griffith's work is primarily concerned with the period of time from the Islamic conquests in the Levant and Near East to the Mongol Conquests of the 13th century. He is also more focused on Egypt, the Levant, and Mesopotamia than Iran itself, but he does have cause to mention it on occasion. He does note however that in Mesopotamia and in the other former border regions between the Byzantines (or Eastern Romans if you prefer) and the Persians, certain Eastern Christian communities did not really take hold, namely the Melkites who were commonly seen as being particularly aligned with Byzantine Christianity over the Eastern Churches such as the Coptic Church or the Syriac Church.

Griffith points to two moments of particular importance in the decline of Christian populations in the Near East in the time frame that he discusses, in the immediate aftermath of the Crusades and the Mongol conquests. The immediate aftermath of the Crusades left much of the local Christian populations vulnerable to reprisals from the Islamic polities still surviving. While the Oriental Churches had not been particularly embraced by the Crusaders they were still tarred by association.

The second factor was the Mongol invasions of the 13th century. Griffith points specifically to the abandonment of Arabic by Christians as the language of expression for theology as indicative of this trend, and cites the anti-Christian policies that followed under the guidance of figures like Ibn Taymiyyah. He does note however that some Churches, such as the Armenian Church, never adopted Arabic as the language of theological debate or philosophical expression, and he is far more focused on the experience of Christians under Arabic speaking Muslims who adopted Arabic for their writings. He posits that following the Mongol invasions this tradition of using Arabic as a language for Christian theology fell by the wayside rapidly. He doesn't connect this to an explicit reason for Christian decline, just that it paralleled rising anti-Christian sentiment following the Mongol conquests.

Given this history I think it is unfair to place a great deal of blame specifically on Timur as the decline of Christian populations in Islamic dominated areas was long standing and punctuated by bouts of more severe restriction and persecution that predated his rule.

I'm interested by this question because I actually didn't realize this was a thing! It's not come up for me from the Timurid studies perspective - this must be a narrative coming out of Syriac studies. So the short answer I would give to this, given the limitations of my access to Syriac sources, is definitely not. I suspect there was a general pattern of decline, and as is usual, a particular person or event gets the blame.

Beatrice Forbes Manz's Rise and Rule of Tamerlane is still the best overall treatment of Timur as a person and a ruler, although it's a little dry, and she comes to the conclusion that Timur is a religious opportunist with a fairly flexible style in general - he's interested in and curious about religion, he likes to use it to further his aims, he's got personal loyalties to the Naqshbandi Sufi order and the family of the prophet but he's not going to let it get in the way of anything. It's hard to even give him a definitive label of Sunni or Shi'ite, although that's period-typical. He wasn't a fanatic for anything but his own (and, as a symbol of himself, Samarqand's) power and glory.

I find this fundamentally convincing. There are plenty of reports from people who met him, like Ibn Khaldun that show an interest in intellectual debate on many topics, including religion, and a willingness to let people disagree if they're clever about it. He liked to pick up artists and scholars to take them home - against their will, but it gives you a sense of what he valued. All of this is to say: mass-conversion is simply not Timur's particular interest area. Even the Cambridge History of Iran chapter on him by H.R. Roemer, which is pretty negative overall, doesn't suggest a fundamentally different assessment of his character. I just don't buy that he was engaging in the kind of intentional global policies that would result in the extermination of a religious community.

Edit: I can't believe I forgot to mention Clavijo!! The travelogue of Ruy Gonzales de Clavijo is truly priceless for getting a first-hand image of what the environment of Samarqand and Timur's immediate circle was like, and he mentions a) Armenian Christians along his route there who are not only alive but brigands causing trouble to him, and b) tons of different types of Christians (Armenian, Nestorian, etc.) at the court itself. He does also give a received account of Timur slaughtering all Christians in the city of Arsinga (p. 73 in that link) but there are some clear political reasons given for that act, and Clavijo reports it to be currently inhabited by Christians (among others).