Alkoxide metathesis

The answer is yes, it will be in following equilibrium:

$$ce{EtOH + t-BuO- <=>[$K$] EtO- + t-BuOH}$$

$$K = frac{[ce{EtO-}][ce{t-BuOH}]}{[ce{EtOH}][ce{t-BuO-}]} = frac{[ce{EtO-}][ce{H3O+}]}{[ce{EtOH}]} times frac{[ce{t-BuOH}]}{[ce{t-BuO-}][ce{H3O+}]} = frac{K_mathrm{a}(ce{EtOH})}{K_mathrm{a}(ce{t-BuOH})} \ = frac{1.26 times 10^{-16}}{3.16 times 10^{-17}} approx 4$$

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Edit:

Although the individual $mathrm{p}K_mathrm{a}$ values used to determine the $K$ in this equilibrium are calculated for solutions in water (some of those are only available in DMSO other than water), I assumed the ratio might be closer to the real value. However, an empirical conversion method that transforms $mathrm{p}K_mathrm{a}$ values of arbitrary organic compounds from one solvent to the other is introduced in this reference (Ref.1). Unfortunately, the full paper is behind paywall, but its abstract states that:

An empirical conversion method (ECM) that transforms $mathrm{p}K_mathrm{a}$ values of arbitrary organic compounds from one solvent to the other is introduced. We demonstrate the method’s usefulness and performance on $mathrm{p}K_mathrm{a}$ conversions involving water and organic solvents acetonitrile ($ce{MeCN}$), dimethyl sulfoxide ($ce{Me2SO}$), and methanol ($ce{MeOH}$). We focus on the $mathrm{p}K_mathrm{a}$ conversion from the known reference value in water to the other three organic solvents, although such a conversion can also be performed between any pair of the considered solvents. The ECM works with an additive parameter that is specific to a solvent and a molecular family (essentially characterized by a functional group that is titrated). We formally show that the method can be formulated with a single additive parameter, and that the extra multiplicative parameter used in other works is not required. The values of the additive parameter are determined from known $mathrm{p}K_mathrm{a}$ data, and their interpretation is provided on the basis of physicochemical concepts. The data set of known $mathrm{p}K_mathrm{a}$ values is augmented with $mathrm{p}K_mathrm{a}$ values computed with the recently introduced electrostatic transform method, whose validity is demonstrated. For a validation of our method, we consider $mathrm{p}K_mathrm{a}$ conversions for two data sets of titratable compounds. The first data set involves 81 relatively small molecules belonging to 19 different molecular families, with the $mathrm{p}K_mathrm{a}$ data available in all four considered solvents. The second data set involves 76 titratable molecules from 5 additional molecular families. These molecules are typically larger, and their experimental $mathrm{p}K_mathrm{a}$ values are available only in $ce{Me2SO}$ and water. The validation tests show that the agreement between the experimental $mathrm{p}K_mathrm{a}$ data and the ECM predictions is generally good, with absolute errors often on the order of 0.5 $mathrm{pH}$ units. The presence of a few outliers is rationalized, and observed trends with respect to molecular families are discussed.

For example, $mathrm{p}K_mathrm{a}$ values of phenol in water, DMSO, methanol, and acetnitrile are illustrated in following diagram:

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Reference:

1. Emanuele Rossini, Art D. Bochevarov, Ernst Walter Knapp, "Empirical Conversion of $mathrm{p}K_mathrm{a}$ Values between Different Solvents and Interpretation of the Parameters: Application to Water, Acetonitrile, Dimethyl Sulfoxide, and Methanol," ACS Omega 2018, 3(2), 1653–1662 (https://doi.org/10.1021/acsomega.7b01895).