Can I still claim my nephew as a dependent on my taxes if he's been in foster care and in a foster home all year?

See https://www.irs.gov/help/ita/whom-may-i-claim-as-a-dependent

No. 

There are two types of dependents, "Qualifying Children"(QC) and Other ("Qualifying Relative" in IRS parlance even though they don't have to actually be related). There is no income limit for a QC but there is an age limit, student status, a relationship test and residence test. Only a QC qualifies a taxpayer for the Earned Income Credit and the Child Tax Credit. They are interrelated but the rules are different for each.

The support test is different for each type. The support test, for a QC, is only that the child didn't provide more than half his own support. The support test for a Qualifying Relative is that the taxpayer provided more than half the relative's support.

He cannot be your QC because he did not live with you.   He cannot be your Other Dependent because you did not (most likely) provide more than half his support for the year. 

The  Foster parents will most likely be claiming him.   A legally place foster child is considered related for tax purposes. 

No.  As @Hal_Al states there are two paths to qualify as a dependent.  The person must either be your qualifying child or qualifying relative.  According to IRS publication 501 outlining the rules for a qualifying child or qualifying relative extracted below, since your nephew did not live with you all year he does not meet the qualifying relative rules nor likely the 1/2 support rule.  

Also since he did not live with you more than 1/2 the year, so he does not meet the qualifying child rules.      

To be a qualifying child: 

• The child must be your son, daughter, stepchild, foster child, brother, sister, half brother, half sister, stepbrother, stepsister, or a descendant of any of them.
• The child must be (a) under age 19 at the end of the year and younger than you (or your spouse if filing jointly), (b) under age 24 at the end of the year, a student, and younger than you (or your spouse if filing jointly), or (c) any age if permanently and totally disabled.
• The child must have lived with you for more than half of the year.
• The child must not have provided more than half of his or her own support for the year.
• The child must not be filing a joint return for the year (unless that joint return is filed only to claim a refund of withheld income tax or estimated tax paid).

To be a qualifying relative:

• The person can't be your qualifying child or the qualifying child of any other taxpayer.
• The person either (a) must be related to you in one of the ways listed under Relatives who don't have to live with you , or (b) must live with you all year as a member of your household (and your relationship must not violate local law).
• The person's gross income for the year must be less than $4,400.
• You must provide more than half of the person's total support for the year.

For more information please see the TurboTax Help article Who can I claim as my dependent? to see all of the factors that may qualify him.