So, if I have a stacked up Archmage Ascension and decide to play out a Spirit of the Labyrinth, would that limit my ability to tutor with Archmage Ascension? The way I read it, AA lets me tutor in lieu of drawing, so since the draw never happens, I should be able to use AA to get around SotL affecting me, right?
If Spirit of the Labyrinth keeps you from drawing a card, Archmage Ascension's second ability can't replace that with searching your library and its other effects instead. In general, if something is impossible, it can't be replaced (C.R. 614.7, 101.2, 101.3; see also C.R. 614.17c).
If Spirit of the Labyrinth is on the battlefield, Archmage Ascension has six or more quest counters on it, and you haven't drawn a card yet this turn, and—
if you would draw one card, then you can still choose to search your library, etc., instead. Once you do draw a card, however, it would be impossible for you to draw any further card in the same turn (while Spirit of the Labyrinth is still on the battlefield), so Archmage Ascension's replacement effect won't be invoked further.
if you would draw more than one card at once—
—as a cost or as a choice (for example, Consecrated Sphinx or Sylvan Library [C.R. 118.12]), then you can't pay that cost or make that choice (C.R. 121.2b).
—other than as a cost or choice, then you draw them one at a time (C.R. 121.2). You can apply the replacement effect of searching your library, etc., multiple times this way (C.R. 121.2b), but once you choose not to and actually draw a card, any further card draws you would do would be impossible, so they can't be replaced and are ignored (C.R. 614.7, 121.6b, 101.2, 101.3; see also C.R. 614.17c).
For example, if Spirit of the Labyrinth is on the battlefield, Archmage Ascension has six or more quest counters on it, you haven't drawn a card yet this turn, and an effect makes you draw three cards other than as a cost or choice, you have the following choices:
Apply Archmage Ascension's effect three times. You will not have drawn any cards.
Apply Archmage Ascension's effect twice, then draw the third card.
Apply Archmage Ascension's effect once, then draw the second card. The third card can't be drawn, so nothing further happens.
Draw one card. The second and third cards can't be drawn, so nothing further happens.
Note that you must have drawn two or more cards this turn (and not had them replaced with something else) before the end step begins for Archmage Ascension's first ability to trigger (C.R. 603.4, 603.2).
EDIT (May 4): Added citation to a rule added in Amonkhet.
EDIT (Aug. 5): Added rule citation.
EDIT (Sep. 4): Edited to conform to rule changes in Commander 2017.
EDIT (Sep. 28): One rule was renumbered with Ixalan.
EDIT (Apr. 4, 2018; Apr. 21, 2018): Correctness edit.
EDIT (Sep. 29, 2018; Dec. 29, 2018; Mar. 22, 2019): Added rule citation.
EDIT (Nov. 21, 2019): Edited, including because some rules were renumbered with Core Set 2020.
If Spirit of the Labyrinth is on the battlefield, Archmage Ascension has six or more quest counters on it, and you haven't drawn a card yet this turn, and—
—other than as a cost or choice, then you draw them one at a time (C.R. 121.2). You can apply the replacement effect of searching your library, etc., multiple times this way (C.R. 121.2b), but once you choose not to and actually draw a card, any further card draws you would do would be impossible, so they can't be replaced and are ignored (C.R. 614.7, 121.6b, 101.2, 101.3; see also C.R. 614.17c).
Note that you must have drawn two or more cards this turn (and not had them replaced with something else) before the end step begins for Archmage Ascension's first ability to trigger (C.R. 603.4, 603.2).
EDIT (May 4): Added citation to a rule added in Amonkhet.
EDIT (Aug. 5): Added rule citation.
EDIT (Sep. 4): Edited to conform to rule changes in Commander 2017.
EDIT (Sep. 28): One rule was renumbered with Ixalan.
EDIT (Apr. 4, 2018; Apr. 21, 2018): Correctness edit.
EDIT (Sep. 29, 2018; Dec. 29, 2018; Mar. 22, 2019): Added rule citation.
EDIT (Nov. 21, 2019): Edited, including because some rules were renumbered with Core Set 2020.