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-MadMage
The player won't be able to draw a card if they already drew a card this turn, assuming one of that player's opponents controls Narset.
Sword of Fire and Ice will deal damage to the targeted creature or planeswalker (or player) (C.R. 115.4), but even if that creature now has lethal damage or that planeswalker now has no more loyalty counters, that permanent will remain on the battlefield long enough to affect whether or how a card is drawn with Sword of Fire and Ice. (This applies to Leovold, Emissary of Trest, Notion Thief, and Hullbreacher just as it does to Narset, Parter of Veils.) Only after Sword of Fire and Ice's triggered ability finishes resolving is—
that creature destroyed, if that creature has lethal damage (C.R. 704.5g), or
that planeswalker sent to the graveyard, if that planeswalker has no more loyalty counters on it (C.R. 704.5i, 306.5c; see also C.R. 306.5),
as the case may be (C.R. 117.2e, 117.3b, 117.5). Remember that it's not damage, by itself, that destroys (not: kills) creatures, but rather a state-based action could do so (C.R. 120.5, 704.5g-h).
Compare this to the case when a spell or ability destroys the permanent (or otherwise removes the permanent from the battlefield), then has a player draw a card (as with Repeal or the last abilities of Teferi, Time Raveler or Ob Nixilis, the Hate-Twisted). In the case of Narset, Narset's first ability's effect will end before that card is drawn (C.R. 611.3b, 611.3, 113.6, 604.1), so that Narset won't change whether or how that card is drawn. This applies similarly to Leovold, Notion Thief, and Hullbreacher. See also C.R. 608.2c and 608.2f, and see also this thread and this thread.
Using card tags is mandatory in the Magic Rulings forum. Please check your private messages to learn how to use them. I've added them for this post.
-MadMage
Sword of Fire and Ice will deal damage to the targeted creature or planeswalker (or player) (C.R. 115.4), but even if that creature now has lethal damage or that planeswalker now has no more loyalty counters, that permanent will remain on the battlefield long enough to affect whether or how a card is drawn with Sword of Fire and Ice. (This applies to Leovold, Emissary of Trest, Notion Thief, and Hullbreacher just as it does to Narset, Parter of Veils.) Only after Sword of Fire and Ice's triggered ability finishes resolving is—
Compare this to the case when a spell or ability destroys the permanent (or otherwise removes the permanent from the battlefield), then has a player draw a card (as with Repeal or the last abilities of Teferi, Time Raveler or Ob Nixilis, the Hate-Twisted). In the case of Narset, Narset's first ability's effect will end before that card is drawn (C.R. 611.3b, 611.3, 113.6, 604.1), so that Narset won't change whether or how that card is drawn. This applies similarly to Leovold, Notion Thief, and Hullbreacher. See also C.R. 608.2c and 608.2f, and see also this thread and this thread.