The first law of  thermodynamic has the following limitations:

(a) The first law does not indicate the direction in which the change car occur- 

For examples, (i) When two bodies at different tempeatures are put in thermal contact with each other, heat flows the body at higher temperature to the body at lower temperature. We now know that heat cannot flow the body at lower temperature to the body at higher temperature,even if first law of thermodynamic is not violated.

(ii) When a moving car is stopped by applying breaks, work done against friction is converted into heat. When the car cools down, it does not start moving with the conversion of all its heat energy into mechanical work. 

(b) The law given no idea about the extent of change.

(i) Our observation and experience tell that there appears to be no restriction conversion of mechanical work into heat.But, there are severe restrictions on the reverse process, i.e., Conversion of heat energy into mechanical energy.

(ii) We know that heat is not converted into mechanical energy all by itself. An external agency called the heat engine is required for the purpose.

(iii) No heat engine can convert all the heat energy received from the source into mechanical energy. The first law of thermodynamics is silent about all this.

(c) The first law of thermodynamics given no information about the source of heat i.e., whether it is a hot or a cold body.