How many elves does Santa need to deliver gifts?

JavaScript (ES6), 52 bytes

s=>eval(s.replace(/\D|$/g,m=>`.0*${m=="*"?3:2}+`)+0)

Explanation

Converts the input into a valid JavaScript statement. Replaces all * with .0*3+ and all other (non-digit) symbols with .0*2+. For example 8*9+*10 becomes 8.0*3+9.0*2+.0*3+10. Finally it appends .0*2 to the end for the last nice count. This works because n.0 = n and .0 = 0.

s=>
  eval(                      // execute the formed equation
    s.replace(/\D|$/g,       // replace each symbol (and also add to the end) with:
      m=>`.0*${m=="*"?3:2}+` // case * = ".0*3+", else replace with ".0*2+"
    )
    +0                       // add "0" to the end for the trailing "+"
  )

Test

var solution = s=>eval(s.replace(/\D|$/g,m=>`.0*${m=="*"?3:2}+`)+0)

<textarea id="input" rows="6" cols="30">1*2+*+*4+1*
2*4+3*+1*6+*
*+*+4*2+1*1
*4+*3+1*+2*3
3*10+2*+*5+*</textarea><br />
<button onclick="result.textContent=solution(input.value)">Go</button>
<pre id="result"></pre>

Flex+C, 112 90 bytes

 m=3,t;
%%
[0-9]+ t+=m*atoi(yytext);
\* m=2;
[+\n] m=3;
%%
main(){yylex();printf("%d",t);}

The first character is a space. Compile with:

flex -o santa.c santa.l
cc santa.c -o santa -ll

Reads from STDIN, writes to STDOUT. Input is terminated by EOF (Ctrl+D in console).

Pyth, 21 bytes

ssMs*VCcR\*scR\+.z_S3

Multi-line example

Single-line test suite

ssMs*VCcR\*scR\+.z_S3
                .z        Take a input, as a list of lines.
            cR\+          Chop each line on '+'.
           s              Flatten into list of strings.
       cR\*               Chop each line on '*'.
      C                   Transpose, into a list of naughty and nice.
    *V            _S3     Vectorized multiplication with [3, 2, 1]. This replicates
                          the naughty list 3 times and the nice list 2 times.
   s                      Flatten.
 sM                       Convert each string to an int.
s                         Sum.