Lorentz Force on a Current Carrying Wire

Newton's third law in modern terms states conservation of momentum. Electrostatic forced conserve $P_{kin}=sum_i m_i p_i$ but magnetic forces do not , as argued by @Rohit. In the presence of electromagnetic fields the conserved momentum is $P = P_{kin} + P_{pot}$, where $P_{pot} = - sum_i q_i vec A_i$.

Yes this is true, for example Lets take 2 parallel infintite wires carrying a current $I_1$ and $I_2$ that located distance $d$ from each other

We know from biot savart law that magnetic field of wire at distance $d$ is $$frac{mu_0 I}{2pi d}hat phi$$ so now we can calculate to Force of wire 1 on wire 2 by your formula $$vec{F} = I int vec{d}ell times vec{B}$$ and getting the result$$F=I_{2}left(frac{mu_{0} I_{1}}{2 pi d}right) int d ell_{2}$$ and the force per unit lengh will be $$f=frac{mu_{0} I_{1} I_{2}}{2 pi d}$$

Now you can see that if $I_1$ and $I_2$ are in the same direction we getting repeling force, and if $I_1$ and $I_2$ are in opposite directions we getting attractive force.

Of course we could do the same by calculate the force of wire 2 on wire 1 and getting the same result.