Yes.

Favored weapons are relevant for certain feats, but they are never a requirement to initiate a maneuver. Any requirements for initiating a maneuver, other than knowing it and having it available (rather than expended) will be explicitly listed in the maneuver's entry in ToB; if you don't see a required weapon, there isn't one.

Each Maneuver Must Be Evaluated Individually...

There's no way to generalize the applicability of every maneuver to every ability that grants additional attacks. Some can be used and some can't.

In your example, while the 1st-level Desert Wind martial maneuver burning blade (boost) (ToB 52) does last, like its duration says, until "End of turn," but the 2nd-level Stone Dragon martial maneuver mountain hammer (strike) (ToB 84) says, "As part of this maneuver, you make a single melee attack." That really means only one single attack gets the benefits of the martial maneuver mountain hammer, even if the target's dropped by a creature using the martial maneuver mountain hammer who has the feat Cleave (PH 92). That's because buried in the Blade Magic chapter in the section (Type) under the heading Strike there's this text:

Because strikes allow for a specific form of attack, you cannot benefit from spells or effects that grant you extra attacks when making a strike (such as the haste spell or a speed weapon). (43)

Thus an adept possessing the feat Cleave who initiates the martial maneuver mountain hammer who drops a foe does get to make the attack granted via the feat Cleave, but he gains no benefit from the martial maneuver mountain hammer when he does. To be clear, when he's granted that extra attack from the feat Cleave, the adept isn't benefiting from "extra attacks when making a strike" but benefiting from extra attacks after resolving the strike--that is, as a result of the strike.

...But Many Boosts Last Until the Turn Ends

In the heading labeled (Type) the section Boost reads, in part,

A boost is a maneuver that grants a bonus, often on attack rolls or damage rolls, for the duration of your turn. A boost always requires a swift action, usually allowing you to initiate it before unleashing a standard action or a full attack. Some boosts impart additional effects, such as stun or fatigue, to your attacks, and others provide some additional effect on an enemy you have just successfully struck in battle. If a boost affects your attacks, it applies to all of your attacks for the round in which it was initiated, but its effect ends at the end of your turn. A boost’s effect applies for its duration, no matter which weapon you might wield in that round. Even if you switch weapons in the middle of your turn, the effect of the boost applies to your new weapon as readily as the previous weapon. Each maneuver’s description gives you the details of each boost’s effect. (ToB 42)

Emphasis mine. Therefore the martial maneuver burning blade lasts until the end of the initiator's turn because it affects attacks, but the duration of 6th-level Diamond Mind martial maneuver moment of alacrity (boost) (ToB 64) remains instantaneous.