What is the fate of NADH produced in the liver during oxidation of lactic acid?
The “best guess” in this question is incorrect and the question itself indicates a lack of understanding of the roles of NAD+ and NADH in energy metabolism. (To rectifiy this, Chapters 17 and 18 of Berg et al. are suggested.)

The production of NADH in the oxidation of carbohydrates and fats is the energetic rationale for these processes. Under aerobic conditions the re-oxidation of NADH to NAD+ via the electron transport chain in the mitochondria* generates ATP for the energetic processes of the cell.

The fate of NADH produced by the oxidation of lactate reaching the liver from the blood would be similar under conditions where it is not needed for gluconeogenesis (in the reversal of the GAPDH reaction).  In these circumstances it would be reoxidized to NAD+, generating ATP in the mitochondria*.

[Note also that pyruvate is most likely to be oxidized by the liver mitochondria to produce metabolic intermediates and ATP, rather than transferred to the blood. NADH is certainly not transferred to the blood.] 

*Advanced Point: Cytoplasmic and Mitochondrial NAD

The statement above, that NADH is “reoxidized to NAD+, generating ATP in the mitochondria”, is correct, but may be taken to imply that the NADH enters the mitochondria. This is not the case becase the NADH and NAD+ cannot pass through the mitochondrial membrane (as @tomd has commented). However the electrons that represent the reduced state of NADH do pass through the membrane. They do this in the guise of other molecules which are reduced in the cytoplasm by NADH in what are known as electron shuttles. The electrons enter the electron transport chain and are finally accepted by molecular oxygen. More detail of shuttles can be found in section 18.5 of Berg et al.