Could a Swordsage become a Lich, using the standard Lich template in the MM? The description says "undead spellcaster," but the template itself does not specify any requirements with regards to spellcasting.
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The lich template requires that you create a phylactery (see "The Lich's Phylactery" in the d20 SRD entry for Lich).
The phylactery requires Craft Wondrous Items, the ability to cast spells, and a caster level of 11th or higher.
A swordsage cannot cast spells and does not have a caster level, and therefore cannot create the phylactery and cannot become a lich.
While you're right that the template doesn't mention the necessity of spellcasting, creating the lich's phylactery does, so a swordsage lich exists...
Only via Adaptation as a Single-classed Character
In the Swordsage section under the heading Adaptation there's the following text
But these aren't rules. This isn't even a variant like those from Unearthed Arcana. It reads like Here's an idea! from the author, and that idea isn't formulated very carefully. Going solely by the vague description above the magical swordsage has the potential to break many campaigns, especially if given free reign to pick his own spells from all the game's published spells, even if those spells are only the Range: Touch or Range: Personal spells from the abjuration, evocation, and transmutation schools.
That said, there is literally no more official information about the magical swordsage. If magical swordsages exists in the campaign in which you're playing, the DM crafted the specifics (and, one hopes, crafted them carefully or already has in place a strong gentlemen's agreement), making the whole endeavor house rules.
But Opportunities Abound for Playing an Undead Swordsage
Many undead templates allow the creature to keep its class abilities (e.g. vampire), and other undead can clearly have class levels (e.g. mummy). A swordsage could, through a convoluted series of events, end up as a sepulchral thief (Ci 133-6), which is like a lich but available to sneaky non-spellcasters.
What Do You Want From This?
There is a particularly unformed swordsage 'variant' presented as a suggestion in ToB called the 'Arcane Swordsage', who replaces maneuvers with spells. Given the fact there is almost no detail on that whatsoever, you could rule that to have a Caster Level, and therefore take Craft Wondrous Item, and at level 11, perform the ceremony to become a Lich.
I'm not sure why you want to, though? This seems like the sort of thing that is best handled by fiat or houserules for story reasons.
By RAW there is no way to make a pure swordsage become a Lich, as the Arcane Swordsage has no definite rules whatsoever.
A Swordsage could dip a casting class, take Practiced Caster three times, and take the appropriate feat, but at that point, why not just become a Necropolitan or a Grave Ghoul or hell, a Sword Wraith?
If you want an undead oriental warrior who keeps 'coming back', get the DM to okay that specifically. Liches have a lot of baggage. Undead returning warrior sounds like LA+2 to me. Give him a funky paralyzing gaze attack and call it LA 3.