It used to work when the Cp was defined as (Cor) for the integral (A).

But here the inner integration over tx is trivial because tx is not a variable of the integrand

In[18]:= FreeQ[infectable[ti] Cor[te, Di, Dv, D0], tx]
Out[18]= True

so one could do as well

In[23]:= A1 = 
 FullSimplify[(Inf/N/D0)*phi*tau*
   Integrate[
       Integrate[(ti + Di - te - Dv) infectable[ti] Cor[te, Di, Dv,  D0], {ti, te + Dv - Di, te + Dv}], 
    {te, -Dv, D0 - Dv + Di}]]
Out[23]= (Inf phi tau (Di (2 D0 Di^2 + (3 Di - 2 Dv) (Di - Dv) Dv) + 
   2 Dv ((-Di + Dv)^3 Log[1 - Di/Dv] + Di^3 Log[Dv/Di])))/(4 D0 Dv N)

in the more general case this is also true

In[28]:= FreeQ[infectable[ti] Cp[te, Di, Dv, D0], tx]
Out[28]= True

but the observation does not bring anything. How did you check the consistency (lack of contradictions) of this:

Cp[te_, Di_, Dv_, D0_] := 
 Which[Di + D0 <= Dv + te || Dv + te <= 0,                                      0, 
  D0 < Dv + te && D0 + te < 0 && Di > Dv + te,                                  Inf/(Inf + Ip), 
  D0 < Dv + te && D0 + te >= 0 && Di > Dv + te,                                 (D0 Inf)/(D0 Inf - Ip te), 
  Di <= Dv + te && ((te == 0 && D0 >= Dv) || (D0 >= Dv + te && te >= 0)),       Di/Dv, 
  D0 >= Dv + te && D0 + te < 0 && Di <= Dv + te,                                (Di Inf)/(D0 Ip + Inf (Dv + te)), 
  D0 >= Dv + te && te < 0 && D0 + te >= 0 && Di <= Dv + te,                     (Di Inf)/(Dv Inf + Inf te - Ip te), 
  D0 < Dv + te && D0 + Di >= Dv + te && D0 + te < 0 &&  Di <= Dv + te,          (Inf (D0 + Di - Dv - te))/(D0 (Inf + Ip)), 
  D0 < Dv + te && D0 + Di >= Dv + te && te < 0 && D0 + te >= 0 && Di <= Dv + te,(Inf (D0 + Di - Dv - te))/(D0 Inf - Ip te), 
  te == 0 && D0 < Dv && D0 + Di >= Dv,                                          (D0 + Di - Dv)/D0,  
  D0 < Dv + te && D0 + Di >= Dv + te && te >= 0,                                1 + (Di - Dv)/(D0 - te), 
  D0 >= Dv + te && D0 + te < 0 && Di > Dv + te &&  Dv + te > 0,                 (Inf (Dv + te))/(D0 Ip + Inf (Dv + te)), 
  D0 >= Dv + te && D0 + te >= 0 && Di > Dv + te &&  Dv + te > 0,                (Inf (Dv + te))/(Dv Inf + (Inf - Ip) te), 
  True,                                                                         0
]

if the default evaluates to 0, then the first case is useless, because it also evaluates to 0.

Have you tried to reduce that using your Assumptions:

{$Assumptions = 
   Di < Dv && Dv < D0 && Di > 0 && Di \[Element] Reals && 
    D0 \[Element] Reals && Inf > 0 && Inf \[Element] Reals && Ip > 0 &&
     Ip \[Element] Reals};

there you state Di < Dv < D0 and in Cp you consider te == 0 && D0 < Dv. Di, Dv, and D0 seem to be some constants. te is the dynamic variable. Despite the fact that Inf and Ip remain undefined, plots like

Plot[Cp[x,1,2,4], {x, -1, -5}]

do give a continous function. That suggests that most cases in Cp are somehow useless in the Di < Dv < D0 range and possibly after simplification you could reach a result. If all that Inf and Ip related stuff is kicked out in a mood of jazz it came to

In[56]:= Cp[te_, Di_, Dv_, D0_] := 
 Which[Di <= Dv + te && ((te == 0 && D0 >= Dv) || (D0 >= Dv + te && te >= 0)),  Di/Dv,
  te == 0 && D0 < Dv && D0 + Di >= Dv, (D0 + Di - Dv)/D0, 
  D0 < Dv + te && D0 + Di >= Dv + te && te >= 0, 1 + (Di - Dv)/(D0 - te), 
  True, 0]

In[57]:= infectable[t_] := UnitStep[t]*UnitStep[D0 - t]

In[58]:= {$Assumptions = 
   Di < Dv && Dv < D0 && Di > 0 && Di \[Element] Reals && 
    D0 \[Element] Reals && Inf > 0 && Inf \[Element] Reals && Ip > 0 &&
     Ip \[Element] Reals};

In[59]:= A = 
 FullSimplify[(Inf/N/D0)*phi*tau*
   Integrate[
    Integrate[
     Integrate[
      infectable[ti] Cp[te, Di, Dv, D0], {tx, te + Dv, ti + Di}], {ti,
       te + Dv - Di, te + Dv}], {te, -Dv, D0 - Dv + Di}]]

Out[59]= (Inf phi tau (Di (6 D0 Di^2 + 
      Dv (5 Di^2 - 15 Di Dv + 6 Dv^2)) - 
   12 (Di - Dv)^3 Dv ArcTanh[Di/(Di - 2 Dv)]))/(12 D0 Dv N)

You're welcome. The specific setting where Do>Dv>Di and te<0

In[7]:= (* First setting: Do>Dv>Di and te<0 *)
Clear[Cp, infectable]
Cp[te_,Di_,Dv_,D0_]:=Which[D0>=Dv+te&&te<0&&D0+te>=0&&Di<=Dv+te,(Di Inf)/(Dv Inf+Inf te-Ip te),D0<Dv+te&&D0+Di>=Dv+te&&te<0&&D0+te>=0&&Di<=Dv+te,(Inf (D0+Di-Dv-te))/(D0 Inf-Ip te),D0>=Dv+te&&D0+te>=0&&Di>Dv+te&&Dv+te>0,(Inf (Dv+te))/(Dv Inf+(Inf-Ip) te),True,0]
infectable[t_]:=UnitStep[t]*UnitStep[D0-t]
In[6]-= {$Assumptions=Di<Dv&&Dv<D0&&Di>0&&Di\[Element]Reals&&D0\[Element]Reals&&Inf>0&&Inf\[Element]Reals&&Ip>0&&Ip\[Element]Reals};
In[10]:= A=FullSimplify[(Inf/N/D0)*phi*tau*Integrate[Integrate[Integrate[infectable[ti] Cp[te,Di,Dv,D0],{tx,te+Dv,ti+Di}],{ti,te+Dv-Di,te+Dv}],{te,-Dv,D0-Dv+Di}]]

has a result (shown as picture because of the formatting )

because of the factor in the denominator (Inf - Ip)^4 it's clear that a condition appears. That's solved by Mathematica. If one uses one Integrate only, one integral remains.

In the second case for the setting where Do > Di > Dv and te < 0 the conditional expression under the remaining integral works into your hands stating

-Inf (Dv + te)^2 (-2 Di + Dv + te)/(2 (Dv Inf + (Inf-Ip) te)) for D0>= Dv + te && Di > Dv + te && Dv + te > 0
0 elsewhere

because you say te < 0 the upper integration limitD0 +Di -Dv for te must be smaller than 0

there is a contradiction: If Di > Dv and D0 > Dv then D0 + Di - Dv > 0 so te > 0 (not te < 0) on the upper limit .... okay, let's send this back to you and continue with the integration limits:

The lower limit is fullfilled: D0 > 0 && Di > 0 && 0 >= 0 because D0> Dv in the assumptions and if te < 0 at least on the lower limit one has Dv > 0.

The upper limit D0 >= D0 + Di && Di > D0 + Di && D0 + Di > 0 is not fullfilled because Di > 0; one has to determine where the integrand is 0 and where not. To get out of this mess one could plot it

With[{Dv = 1, Di = 4.5, D0 = 8},
 Plot[If[D0 >= Dv + te && Di > Dv + te && Dv + te > 0, 
   1, -1], {te, -Dv, D0 + Di + Dv}]
 ]

fortunately there is only one switch from 1 to -1 at te = Di - Dv leaving us behind with the integral

In[30]:= Clear[Dv, Di, D0]
Inf phi tau Integrate[-((Inf (Dv + te)^2 (-2 Di + Dv + te))/(
    2 (Dv Inf + (Inf - Ip) te))), {te, -Dv, Di - Dv}, 
   Assumptions -> Dv > 0]/(D0 N)

Out[31]= ConditionalExpression[-((
  Inf^2 phi tau (-Di (Inf - Ip) (4 Di^2 (Inf - Ip)^2 - 6 Dv^2 Ip^2 + 
        9 Di Dv Ip (-Inf + Ip)) + 
     6 Dv^2 Ip^2 (2 Di (Inf - Ip) + Dv Ip) Log[(Dv Ip)/(
       Di Inf - Di Ip + Dv Ip)]))/(12 D0 (Inf - Ip)^4 N)), 
 Di Ip^2 < Di Inf Ip || Di (Inf - Ip) (Di (Inf - Ip) + Dv Ip) < 0]

the integrals are done. Please check for errors.