There are situations in which one starts off the Omer not being obligated in counting but at some point during the Omer becomes obligated. A classic example is a boy whose Bar Mitzvah falls during the Omer; previous to that point he was a minor and was not obligated but then becomes obligated afterwards. Other examples include someone who converts to Judaism in the middle of the Omer, or someone who has a surgical procedure and cannot count for one or more days. The main issue is, since one opinion holds that the entire forty nine-day period is one continuous Mitzvah, if one misses one day of counting the Mitzvah is disrupted and thus one may no longer continue counting with a blessing.
The Sha’are Teshuva (O.H. 489:20), quoting the HIDA (Birke Yosef § 489), rules that one who becomes a Bar Mitzvah or who converts during the Omer, cannot continue counting with a blessing. They may, and should, continue counting without a blessing. This is consistent with the HIDA, who even posits that if one misses a day of counting, the previous days that one did count with a blessing may be considered a Beracha Levatala, a blessing in vain. However, the Sha’arei Teshuva explains that nowadays, many pre-Bar Mitzvah boys count the Omer with a blessing so as to train themselves in the Mitzvah and that there is a certain measure of importance even to these blessings. As such, in such a case, a boy who reaches Bar Mitzvah during the Omer would be permitted to continue counting with a blessing.
Rabbi Ovadia Yosef (Yabia Omer, vol. III, § 28-29) disagrees with the Sha’are Teshuva’s position and says that a child who becomes Bar Mitzvah during the Omer may not continue counting with a blessing. Nevertheless, Rabbi Ben Zion Abba Shaul (Or LeZion, vol. I, § 36), Rabbi Shlomo Zalman Aurerbach (Halichot Shlomo, ch. 11, § 10), Rabbi Shmuel Wosner, Rabbi Nissim Karelitz and Rabbi Haim Kanievsky all rule that a child may indeed continue counting with a blessing. Unlike a child, who at least had the obligation of training in the Mitzvot, a convert has no obligation whatsoever to count prior to converting, and thus if one became Jewish during the Omer, one would not be permitted to count with a blessing for that year.
Additionally, there is a debate regarding one who knows ahead of time of a surgical procedure and the like which will take place during the Omer during which one would not be able to count. On one hand, one is aware that the Mitzvah will be interrupted at some point and perhaps one should not even begin counting with a blessing, lest it be considered a Beracha Levatala. Nevertheless, Rabbi Ben Zion Abba Shaul says that, although the Mitzvah is described in the Torah as being seven complete weeks, in the case of one who foresees an operation, one’s counting up to that point is consecutive and complete. As such, he permits one in that situation to count each day leading up to the operation with a blessing.
Summary: A boy who becomes Bar Mitzvah during Sefirat HaOmer may count with a blessing from that point on. One who converts during the Omer may not count with a blessing. One who knows that one’s counting will be interrupted at some point may go into the Omer and count with a blessing.