Here is a short simple available answer provided online by Prachi Sawant in 2016 to quote:
Formic acid has both aldehydic (-CHO) and carboxylic (-COOH) functional groups.
Hence, it gives both Tollen's and Fehling's test positive.
Reaction of formic acid with Tollen's reagent:
$\ce{HCOOH + 2[Ag(NH3)2]+ +2OH- → 2Ag + CO2 + 2H2O + 4NH3 }$
Reaction of formic acid with Fehling's reagent:
$\ce{HCOOH + 4OH- + 2Cu^{2+}(in complex) → CO2 + 3H2O + Cu2O }$
Now, here is my attempt on the possible underlying mechanic, which is likely complex. I start with the above net reaction, which in reality, is probably not completely accurate, but nevertheless, may provide some insight into actual half-cell reactions, beginning with the reduction of the organic cupric complex:
$\ce{Cu^{2+} + e^- -> Cu^+ }$
where the insoluble Cu2O (the test indicator) is formed. The corresponding other half-cell is less clear accept for the liberation of an electron:
$\ce{COOH^- -> .HOCO + e^- }$
which may speculatively may be followed by:
$\ce{.HOCO + .HOCO -> H2O + CO + CO2 }$ (speculation)
And/or, this paper favoring non-conventional ionized species in the context of formic acid as possessing more stability as demonstrated, for example, by:
$\ce{HCOOH -> [.HCOOH]^+ + e^- }$
With its source cited below, followed by:
$\ce{[.HCOOH]^+ -> H2O .CO^+ }$ (a distonic ion, per 2nd cited source)
$\ce{OH^- + H2O .CO^+ -> H2O + COOH^- }$ (speculation)
To quote a cited source above :
Rate constants for the radical-induced hydrogen abstraction from formic acid, HCOOH, are presented here. Only those reactions leading to the formation of $\ce{.HOCO}$ were investigated.
Per a source: 'The gas phase chemistry of the formic acid radical cation $\ce{.[HCOOH]^+}$ Mechanism for exchange of the hydrogen atoms: a quantum chemical investigation', to quote:
Earlier labelling experiments had indicated that prior to the loss of H the hydrogen atoms in ionized formic acid, $\ce{.HCOOH^+}$, can become positionally equivalent via an unknown pathway. Using ab initio molecular orbital calculations we have located a minimum energy reaction pathway for hydrogen exchange reactions which takes place close to the dissociation level and proceeds via flexible ion/dipole complexes. Along this path three equilibrium structures of different atom connectivity are found: $\ce{.HCOOH^+}$, 1, ionized formic acid; two forms of the surprisingly stable ion/dipole complex $\ce{.HOHCO^+}$, 2, 3; and the distonic ion $\ce{H2O CO.^+}$, 4. Ion 1 is separated from ion 2 by a barrier of 142 kJ mol−1 and this barrier corresponds to a 2A′ → 2A″ surface crossing. The barrier for the proton transfer 3 → 4 is only 25 kJ mol−1 relative to 3; the transformation $\ce{.HCOOH^+ -> H2O .CO^+}$, formally a 1,2-hydrogen shift, takes place via the ion/dipole complex $\ce{HO· HCO^+}$, $\ce{H2O .C^+}$ is the reacting configuration for decarbonylation and this reaction is calculated to take place at the dissociation limit in full agreement with experiment.
Where ‘decarbonylation’ can also be effected by several transition metal complexes, (source), as in the current context with an organic copper complex.